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yUafh f^f'-il
HARVARD COLLEGE
SCIENCE CENTER
LIBRARY
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10 • ' • . ^^ • '
TABLES
OF
LOGAEITHMS OF NUMBEES,
AND or
LOGARITHMIC SINES, TANGENTS AND SECANTS,
TO SEVEN PLACES OF DECIMALS;
TOGITHEft WITH
OTHER TABLES OF FREQUENT USE
IN THK
STUDY OF MATHEMATICS,
AND IN
PRACTICAL CALCULATIONS.
BY
ANTHONY D. STANLEY, A. M.
ProfeHor of Mathematica in Yale College.
NEW HAVEN:
DURRIE A PECK.
PRINTED BY B. L. HAMLBN,
Printer to Yale College.
Matr, \-i<:^\'i
gMIVAI0 COLLiUC UMMV v
6IFT Of TNI t^
CMU imrifiiTC
Entered, according to Act of Congress, in the jear 1846,
by Ahthoht D. Stahlst,
in the Clerk's office of the District Court of Connecticut
Compoaition and PreMwork bf
Bt Iji HAMLENi
New HftTen.
BtereotTpad by
RICHARD H. HOBBB,
Hartford.
PREFACE.
Th£ following collection of tables has been prepared, chiefly
with a view to meet the wants of college students. Tables of
much less extent would perhaps be regarded by many as suffi-
cient for this purpose : and such indeed are most of the tables in
conunon use. These smaller tables are doubtless suitable enough
for the illustration of i^rinciples and rules of calculation ; but
from the inaccurate results which they usually give in the solu-
tion of [voblems, they enable the mathematical student who em-
ploys them, to see but little of the use of tables to the computer.
Tables in which the logarithms of numbers are carried to as many
as seven places of decimals are needed for the proper solution
of many of the problems to be met with in any common course
of mathematics. Indeed logarithms of five or six places of dec-
imals are now seldom used in calculations requiring any but the
lowest degree of accuracy. In France, at the examinations for
admission to the military and naval schools, candidates have for
several years been required in their computations to use loga-
rithms of seven decimal figures. The tables which are there
most generally used are those of Callet. The tables of like ex-
tent most commonly employed in England are Hutton's: and
those of Vega are most used in Germany. The only similar ta-
bles hitherto published in this country are Hassler's : and these
have been found so inconvenient from the smallness of the type
in which they are printed, that they are much less used than
those of Hutton and Callet imported from abroad!
The following work is one of sufficient extent to furnish nearly
the same aid to computers as is afibrded by the tables of Hutton
or Callet In presenting it to the public, our principal concern
is to show that proper care has been taken to secure its correct-
ness, in carrying it through the press ; as a new work of this kind
inaccurately printed would be worse than useless. We therefore
state the following particulars. The pages of the tables were
first made by the printer to agree very nearly with the copy.
Four or five proofs of each were afterwards read in connection
(4) PREFACE.
with dilSferent tables, before the work was printed. Two and in
several cases three proofs were examined before the page was
stereotyped, and two proofs from the plates were examined after-
wards. This proof reading was done exclusively by the writer.
In comparing the proofs with different tables, the corresponding
columns of figures were in each case placed closely side by side,
that the eye might readily detect any disagreement. An exam-
ination of the proofs with the assistance of another in the ordi-
nary way, would have required much less time : but the other
method was preferred as better suited for the detection of errors.
The tables employed in correcting the proofs were chiefly
those of Tega, Callet, Hutton, Babbage and Shortrede. Babbage
and Callet were most relied on for accuracy. But their tables
were not found free from error. It is a common opinion with
regard to works which have been stereotyped, that while they
are open to correction as errors are discovered in them from time
to time, they are also made permanently secure of whatever de-
gree of accuracy they once acquire. But in using two copies of
Callet's tables, one of recent and one of earlier date, it was seen
that some errors from which the latter was free, had found their
way into the former. A notice of several errors observed in the
different tables which we have had occasion to examine, will be
published in another place.
After the following tables were ready for the press, the editor
found among the Memoirs of the Astronomical Society of Lon-
don an article by Mr. Babbage, in which he mentions six errors
that are common to almost all the tables of logarithms he has
seen. Prom his own table these errors are of course excluded.
And as that had been used in correcting the first proofs of the
following table of logarithms, we were furnished with tolerable
means of forming an opinion concerning the merits or defects of
the first proof reading. On referring to the proofs concerned, it
was found that all the errors in question had been noticed and
corrected. Our second proofs of the table of logarithms were
compared with Shortrede's table. We therefore examined this
table with reference to the errors noticed by Babbage ; and in it
found five of them. We then referred to a list of nine or ten
errors that we had discovered in the same table, and found all of
these five among them ; which encouraged the belief that not
many errors had been overlooked in the second proof reading.
PR£rAC£. (5)
As we are informed that Babbage's table was employed in the
correction of Shortrede's, it seems a little remarkable that all
the five errors referred to should have escaped notice. But the
general correctness of the latter table need not be therefore dis-
trusted, as we may infer from what is stated above, that those
errors form a large proportion of all the errors which it contains.
Next in importance to the table of logarithms of numbers, is
that of logarithmic sines, tangents, &c. Hutton in his table of
sines, tangents, &c, gives them only at intervals of a minute each,
through all except the first two degrees of the quadrant, where
sines and tangents are given for every second. The diflferences
of successive sines, tangents. Sec, in some parts of his table are
accordingly quite large, and in finding intermediate sines, tan-
gents, &c. the process of interpolation is inconveniently long.
There is also room for considerable error in this process, unless
second differences are taken into account, which would render
the work still longer. In Callet's table, sines and tangents are
given at intervals of seconds for the first five degrees, and sines,
cosines, tangents and cotangents, at intervals of ten seconds for
the whole of the quadrant. His table is therefore much more
convenient than Button's. In one respect however it is less con-
venient, as it fails to give secants and cosecants, which Hutton
does not omit.
We have endeavored to combine the main advantages of these
two tables in the second table of this collection. In giving se-
cants and cosecants, our table agrees with that of Hutton. It
differs from his chiefly in giving sines, tangents, &c. at intervals
of ten seconds instead of minutes, for the first fifteen degrees.
In consequence of this and of a special arrangement for the first
two degrees, no error greater than a unit in the last place of deci-
mals can happen in any part of the table, from the neglect of
second differences in interpolation ; whereas there would have
been room for errors thirty-six times as great as this, if the sines,
tangents, &c. had been given, as in Button's table, only to min-
utes, after the second degree of the quadrant.
There are many cases in which a required result cannot be
accurately determined, unless the result be known approximately
from some preliminary calculation. This first calculation does
not conmionly require to be made with much exactness : and if
it be logarithmic, it ought to be made with logarithms of less than
(6) PREFACE.
seven decimal figures. For convenient use in hasty calculations
of this and other kinds, we have added to the larger tables of
logarithms and logarithmic sines, tangents, 6cc, smaller tables of
logarithms, antilogarithms, logarithmic sines, tangents, &c. car-
ried only to five places of decimals.
Before we determined to embrace these tables in the collection
to be made, we had intended that the two leading tables should
be followed by a variety of smaller ones, useful in navigation,
surveying, &c. Several such tables were accordingly selected and
prepared for the press, but most of them were set aside to make
room for the smaller logarithmic tables, which seemed likely to
be of more general use. A few of the most important however
have been retained, and will be found at the end of the volume.
The table on page 338 was copied from one in common use,
the correctness of which we had no reason to distnist. But after
this page was stereotyped, we observed that the numbers in it
could not be depended on for accuracy in their final figures. This
defect may however be deemed of little practical importance, as
no great exactness is called for in cases where those numbers
are used.
The table of meridional parts, which was prepared for the stere*
otyper immediately after page 338, was withheld till it could be
examined, cursorily at least, with reference to such inaccuracies
as had been observed in that page. And in this table too, sev-
eral numbers were found erroneous, by a unit, in the final figure.
These were correctedi;. but the table may still contain errors of
the kind in question, as the examination was a hasty one, being
much less thorough than it would have been had the occasion
for it been earlier suggested. The numbers given in this table
are carried to the first place of decimals. Meridional parts given
in whole numbers, as they commonly are, are unsuitable for use
in many cases where accuracy is demanded.
CONTENTS.
INTBODUCTION.
Page.
Of Logarithms (9)
Of the Table of Logarithms of Numbers . . . • . (16)
" Table of Artificial Sines, Tangents, &c. . . . (27)
" Table of Natural Sines and Cosines . . . (41)
^^ Smaller Table of Logarithms of Numbers . . . (43)
" Table of Antilogarithms (45)
^^ Smaller Table of Artificial Sines, Tangents, dsc. . (46)
« Traverse Table (51)
«* Table of Meridional Rirts (52)
*• Table for correcting Middle Latitude . . (52)
Table of Logarithms for computing Compound Interest, &c. (53)
Tables for interchanging Common and Hyperbolic Loga-
rithms (53)
** Table for finding the Lengths of Circular Arcs . . (54)
'' Tables on page 340 (54)
TABLES.
Table of Logarithms of Numbers 1
Table of Artificial Sines, Tangents, &c 155
Table of Natural Sines and Cosines 279
Smaller Table of Logarithms of Numbers .... 289
Table of Antilogarithms 294
Smaller Table of Artificial Sines, Tangents, &c. . . 300
Traverse Table or Table of Departure and Difierence of Latitude 323
Table of Meridional Parts 330
Workman^s Table for correcting Middle Latitude . . 338
Table of Logarithms for computing Compound Interest, &c. . 339
Table for converting Common into Hyperbolic Logarithms • 339
Table for cotiverting Hyperbolic into Common Logarithms . 339
Table for finding the Lengths of Circular Arcs . . . 339
Table of Logarithms carried to thirty places of decimals . . 340
Table of Numbers often used 340
NoTC. — ^There Ls an error in several copies of the tables, that was occasion-
ed by a mistake of the stereo^rper in substituting for a 6 which had been
nuurked as imperfect, in the final proof for his use. The error is on page 205,
in the cosine ofTSOf 4fy\ which should be 9.99591G9 instead of 9.9959109.
INTRODUCTION.
LOGARITHMS.
Art. 1. AhOQJkBjnnMqfanufnberi8the^xvovExnu}?d(^shews
what power* that nttmber is of a given number. Thus if the
given mimber is 2, the logarithm of 8 is 3, that of 32 is 5, that
of 64 is 6, &c. because 8i=2», 32=2', 64*2% &c. and if 3 is
the given number, the logarithm of 9 is 2, that of 27 is 3, that
of 81 is 4, &c. To distinguish the logarithms of one of these
two classes from those of the other, the former are said to belong
to a system of logarithms of which 2 is the iose, and the
latter to a system of which 3 is the base. To make either
system complete, the logarithms of all the numbers 1, 2, 3, 4,
&c. must be obtained, by regarding each number as a power
of the base, and finding the exponent which denotes what that
power is. Thus if the base is 2, each number is to be considered
as a power of 2, and the exponent denoting the power is to be
ascertained, and taken for the logarithm of the number. For
example, let the number be 13 : this is greater than 2' and less
than 2< ; and there is some number between 3 and 4, which in-
dicates what power 13 is of 2 : that number is the logarithm of
13 ; and though not assignable at once, it may be determined
without difficulty by various methods of calculation which need
not be specified here.
Any number except one may be made the base of a system of
logarithms ; for by giving the number different indices, we may
form powers of it that shall equal any numbers whatever ; but
as all the powers and roots of one are alike, we cannot represent
other numbers by any powers or roots of the number one.
2. Logarithms are employed chiefly to facilitate numerical cal-
culations. And for this purpose it is best to make use of the
* The word pcw€T b often ated to denote a quantity with aoy index whether
fractional or integral, poeitive or negative. In thia senee, «'» a , a are all powers
of a. With this extended meaning the term is employed in these pages.
B
( 10). INTRODUCTION.
logarithms of no more than one system. The method of pre*
paring for use a system of logarithms, is to take the numbers 1,
2, 3, 4, &c. in their natural order, (to a greater or less extent as
occasion demands;) and considehng them all as powers of the
base, to determine their logarithms accordingly, and then arrange
the logarithms together with the numbers in a table. The sys-
tem which has been chosen as most conrenient, is that whose
base is 10. It is for reasons pertaining to the common mode of
reckoning by tensy (and to be stated more explicitly in Art.
17,) that this system is preferred. Logarithms in the system
whose base is 10, are called oonwion logarithms or Briggs'* log-
arithms.
The coMifON loffarUhm of a number then is the exponent
which shows what power that number is af 10.
When the kind of logarithms spoken of is not specified, conv-
num logarithms are always to be understood.
3. The logarithms of the numbers which are exact powers of
10 are at once obWous: thus the logarithms of the numbers
1000, 100, 10, 1, tV, Tiir, ttVit are obviously 3, 2, 1, 0, - 1, -2,
— 3, because those numbers are obviously the following powers
of 10, viz. 10% 10% 10% 10% 10-», 10-«, 10-».
The logarithms of other numbers are fractional, and some cal-
culation is necessary to determine accurately what they are.f
Thus for example, the logarithm of 687 is some number be*
tween two and three^ because 587 is between 100 and 1000, and
the logarithm of 100 is two^ that of 1000, three. By a proper
method of calculation it may be ascertained that the logarithm
of 687 is greater than 2.7686381, and less than 2.7686382 ; and
that it differs less from the former number than from the latter.
* We are indebted to Lord Napier a Scottish mathematiciaa, for the ioTention
of logarithms. The firit logarithmic tablei were publiebed bj him in 1614. His
•yatem of logoritbmt had 3»718 ... for its baae. This system was aseamed on
account of a peculiarity that makes its logarithms a little easier of calculation than
those of other systems. The logarithms of this system are eommonly called
Ji'aperianf but are sometimes called also hyperholie logarithms from certain ra-
markable relations which they have to a curve called the hyperbola. At the sug-
gestion of Briggs a contemporary of Napier, the system in which 10 is the base
was early substituted for the other and has ever since been the one in common use.
t Indeed the logarithms of such nombers are not capable of being expressed in
iWustiona with perfiet exactness; but they can be assigned with any lower degree
of accuracy, and ei^ecially with all the accuracy that is requisite for practical
purposes.
LOOABITHMS. (11)
The logarithm of 687 is theiefore very neariy 2.7686381. We
might assign a number still nearer the true logarithm by pro-
ceeding to eight or more decimals ; but for ordinary purposes it
is si^Scient to know the logarithm of a number accurately to
the seventh decimal place. Accordingly 2.7686381 may in prac*
tice be regarded as the logarithm of 587.
4 Again the logarithm of 0.0357 is between — 1 and •*- 2, be-
cause 0.0357 is less than 0.1 that is 10" *, and greats than 0.01
that is 10'' ; and by due calculation it has been found diat the
logarithm of 0.0357 differs but little from - L4473318.
The logarithm as thus expressed consists of a negative whole
number — 1 and a negative decimsd —.4473318. There are rea-
sons for avoiding the use of negative decimals in logarithms, which
is done by substituting for them such equivalent expressions that
each negative logarithm may be made to consist of a positive
decimal and a negative whole number. Thus in the example be-
fore UB, instead of —.4473318 is substituted the equivalent ex-
pression -1+. 5526682: the whole logarithm thus becomes
- 2+*5526682, consisting of a negative integer -2 and a posi-
tive decimal +.5526682.
To represent a l(^arithm which is thus composed of a negative
integer and a positive decimal, the decimal without its sign +
is placed immediately after the integer, and the sign — placed
wer the latter. Thus the logarithm of 0.0357 is 2.5526682,
which is to be understood as equivalent to the expression
-2 +.5526682.
5. The integral parts of the logarithms of 587 and 0.0357,
viz. 2 and -*2, were easy to be determined ; the Ic^arithm of
587 being obviously more than 2 and less than 3, and the loga-
rithm of 0.0357 being between — 1 and —2. Likewise in every
case the integral part of a logarithm may be readily assigned.
To distinguish the integral from the decimal part, the former is
called the index or more commonly the charaeteristic of the logar
rithm, and the latter the matiHssa, Thus 2 is the characteristic of
the logarithm of 587, and .7686381 the mantissa ; — 2 is the char-
acteristic of the logarithm of 0.0357, and .5526682 the mantissa.
6. The general rule to determine the eharaderistic of a loga-
rithm is as follows.
If the number whose logarithm is in question consists of one
or more integral figures with or without a decimal annexed, the
(12) INTBODUOTION.
characteristic of the Ic^arithm is one less than the number of the
integral figures.
Thus the characteristic of the logarithm of 504.27 is 2, as
the integral figures viz. 504 are three in number. And the char-
acteristic of the logarithm of 2.036 is 0, because the number
2.036 has one integral figure.
But if tte number be only a decknalfractiony the character-
istic of its logarithm is negative, and is greater by one than the
number of ciphers at the beginning of the decimal.
Thus if the number concerned be the decimal .087, the char-
acteristic of its logarithm is ~ 2 or as it should be stated 2 ; and
if the number be .275, the characteristic of its logarithm is 1.
7. In proof of these statements it may be observed that the
least number having n integral figures is lO"*** ; the least num-
ber having 3 integral figures for example being 10' or 100, the
least having 4 such figures being 10' or 1000, &c. and as there
are n integral figures in the number 10*' ^ , the number of such
figures in 10" is n+1. Accordingly if a number has n integral
figures only, it must be less than 10", but cannot be less than
10"" » , though it may be' equal to this. Since then every number
having n integral figures is either 10"" ' or between 10"" ' and 10%
its logarithm must be' either n - 1 or between n — 1 and n, that
is, must be either n — lorn — l+a decimal ; the characteristic
therefore is n ~ 1 ; that is, the characteristic of the logarithm of a
number having integral figures, is one less than the number of
integral figures ; as is stated in the first part of the rule given
above.
Again, the least decimal number beginning with n ciphers is
1
IQ^4, ; the least decimal begmning with one cipher for examjAe
1
being .01 or tq^9 the least begmning with two ciphers being
1 1
.001 or Jq^j &c. and as Tq^+T is a decraml beginning with n
1 . ^ .
ciphers, tw^ is a decmial beginning with n - 1 ciphers. Accord-
ingly, if a decimal begins with n ciphers, it must be less than
jq;» but carmot be less than Jq^t;' though it may be equal to
thi& Since then every decimal beginning with n ciphers is either
LOQAKITHMS. (13)
iO"+"« ^^^ ^® 10"^"+'^, or between i?yi+7 and ^ that is between
10-<»+'>and lO-', its logarithm must either be — (n + 1) or be
between -(n+1) and -n, that is, must be either —(n+1) or
— (n+l)+ adectmal; the characteristic therefore is —(n+1);
so that the characteristic of the logarithm of a decimal is nega-
tive, and is one greater than the number of ciphers at the begin*
aing of the decimal ; as is stated in the second port of the fore-
going rule.
8. When the characteristic of a logarithm is negative, it is
often, for convenience, made positive by the addition of 10;
which can lead to no error, if it be kept in mind that such a
change has taken place, and due care be taken to make allowance
for it afterwards, at a proper time.
Thus instead of the logarithms 1.8272047, 4.0260108, we may
write 9.8272047, 6.0260108.
9. Since we have a rule by which we can readily find the
characteristic of the common logarithm of any proposed number,
tables are needed only to furnish the mantissas of logarithms.
These mantissas have been calculated and arranged in tables to
«uch an extent as has been found convenient for practical pur-
poses. It is unnecessary here to describe the manner of com-
puting them ; it will be sufficient to show how they are arranged
for use in the following tables. In describing these tables how-
ever, it is necessary to refer to certain properties of logarithms
which will now be stated.
10. If Zr, ilf , iV, ^c. be several numbers, and Z, w, n, 4^. their
logarithms in a system of which the base is 6, the logarithm of
LMN 4*c. is equal to f+m+n+ij-c.
For from the nature of logarithms, (Art. 1) L = b\ M^b^j
N^b% Sre, hence LMN ^c. =6'6"6" 4-c.=y+"+"+4^-: but if
LMN^c. is the (Z+w+n+ 4*c.)th power of 6, /+w+n+4^
is the logarithm of LMNSfc. as was asserted.
Therefore the logarithm of the product of several numbers
equals the sum of the logarithms of the numbers,
11. Again, if M and N be two numbers whose logarithms are
m and n, the logarithm of M-^N equals m — n.
For as JIf ^J"* and iV= J% M-t-iV«&*-^'=6"-.
Therefore the logarithm of the quotient of one number divided
by another is equal to the logarithm of the dividend diminished
by thai of the divisor.
(14) INT&ODOCTIOH*
12. Again, if AT is a number vhose logarithm is in, the loga-
rithm of itf is mn.
For as M^lT^ M^^lT* ; bat if iJf* is the mnth power of the
base b, mn is the logarithm of M^.
Therefore the logarithm of any power of a number is obtained
by muUiplying the lofforithm of thai number into the number
denoting the power.
13. Finally, if M is a number whose logarithm is m^ the log-
i. m
arithm of 3f * is -; •
n
For as M=::b'^, JHf^^b"-
Therefore the logarithm of any root of a number is obtained
by dividing the logarithm of that number by the number denoting
the root,
14. The following definition will prepare the way for one or
two inferences to be drawn from these principles.
The significant figures of a number are those which remain
after rejecting all ciphers that occur either at the end or at the
beginning of the number.
Thus the significant figures of 8021700 are 80217; those of
the decimal .00010408 are 10408.
16. If two different numbers have the sam^ significant figures^
occurring in the same order, their logarithms will have different
characteristics, but the sam£ mantissa.
For two such numbers may be made alike, by merely re-
moving the decimal point in one of them to the right or
left; and as changing the position of the decimal point, is
equivalent to multiplying or dividing by 10, 100, 1000, &c.
the greater number must be equal to the smaller one multi-
plied by some exact power of 10, which we will denote by 10",
n being a whole number; then (Art 10) the logarithm of the
greater number must be equal to the logarithm of the smaller
number increased by the logarithm of 10" which is n. Now
by adding the integer n to the smaller logarithm to obtain the
greater, we alter the characteristic, but leave the mantissa un-
changed. The logarithms of the two numbers therefore have
like mantissas with unlike characteristics. Thus the logarithm
of 108.06 is 2+ a decimal, and that of 1080600 is 6+ the same
decimal, because 1080600=:^ 108.06x10^ ; also the logarithm of
L06AR1THHS. (16)
.00203 is -3+ a dedmaly whde that of 20.3 is 1+ the same
decimal,
16. Again, if the logarithms of two numbers have different
characteristics, but the SBtne manUssa, the significant Jigures of
the numbers must be the same, and must ooeur in the same or-
der ; and the numbers will differ only as to the position of the
decimal point.
For then the difference of the logarithms is a whole num-
ber, which we may denote by n ; but this difference is the loga-
rithm of the quotient of one of the numbers divided by the
atbet (Art. 11); ti^ quotient itself therefore is 10*, and accord-
ingly one of the numbers is equal to the other multiplied by
10*; but a number after beii^ multiplied by any exact power
of 10 as 10*, will have the same significant figures as before,
and only the position of the decimal point will be changed. If
for example, the logaritfuns of two numbers are 2+ a decimal
and - 1-V the same decimal, and if it be known that the number
to which the first logarithm belongs is 231.02, then we infer that
the number to which the other logarithm belongs must be
.23102. For the difference of the Ic^arithms is the integer 3,
and the quotient of the greater number divided by the less is
therefore 10* ; but the greater is 231.02, as has be^i supposed ;
therefore the less is 231.02-^ lO^'s. 23102.
17. What is stated in the two preceding articles will apply to
the common logarithms of numbers only. Such logarithms are
therefore much more convenient than others for general use;
since when the common logarithm of any number has been
found, those of various other numbers having the same signifi-
cant figures are obtained without difficulty, simply by changing
the characteristic, according to the princifdes stated above.
(16) INTBODUCTIOH.
Table of Looakithms of Numbebs: pp. 1-163.
18. This is a table of the numbers 1, 8, 3, &c. up to 100000,
with the mantissas of their conunon logarithms, which are also
the mantissas of the logarithms of other munbers whose signiji'
cant figures fall between 1 and 100000, To these the proper
characterisHcs^ omitted for conrenience, are to be prefixed ac-
cording to the rule in Art. 6.
The first three pages contain all the numbers from 1 to 1000,
(except those ending with 0,) arranged in columns headed N.
By the side of each number, in the next column headed ttog. is
to be found the mantissa of its logarithm, carried to seven places
of decimals , the decimal point being omitted, and the first three
figures being separated a little from the other four, for ease of
reading. Thus the true mantissa of the logarithms of 8, 37 and
809, are .9030900, .6682017 and .9079486, respectively.
In the subsequent pages, in columns on the left and headed
Num. are all numbers from 1000 to 10000 ; the first two figures
being stated only at convenient intervals, as they are the same
for several numbers in succession. In the columns headed
0, 1, 2, 3, 4, 6, 6, 7, 8, 9, are the mantissas of the logarithms of
numbers carried to seven places of decimals, or rather the four last
figures only of these mantissas, except in the column headed 0,
where to some of the mantissas, the first three figures also are given.
Each mantissa belongs to a number whose first four figures are op-
posite it in the left-hand column headed Num, and whose fifth
figure stands at the head of the column containing the mantissa.
And the whole logarithm of this number will be the mantissa
with the characteristic 4 prefixed. These mantissas will accordr
ingly be spoken of as the logarithms of the numbers composed
of five figures in the way just mentioned, the characteristic 4
being understood as prefixed to each mantissa, in confoimity
with the rule in Art. 6.
Wherever the character ♦ occurs, a cipher must be understood.
The use of the space between the colunms headed 4 and 6 is,
by dividing the page, to facilitate the finding of any column sought
Also, at the beginning of each page the figures are made heavier,
that as we consult the table they may more readily catch the eye.
TABLE OF LOGABITHMS.
(17)
la the coltunns headed Diff. 4* MuU. are aeveral numbers
placed opposite the nine digits, the number opposite 1 beii^ in
each case the difference of two consecutive logarithms in the
table, and those opposite the other digits being corresponding mul-
tiples of the differences over them.
19. All this will be readily understood from the following in-
dication of the meaning of the first part of page 6.
Nnmben.
LogaritliRU.
Nnmben.
LoguHhnM.
11200
40492180
11212
40496a'tl
11201
4.0492568
• • • «
11202
40492966
11218
40499154
11203
40493343
11219
40499541
11204
40493731
11220
40499929
11205
40494119
11221
40500316
11206
40494506
11222
40500703
11207
40494894
• • • •
11208
40495281
11229
40503411
11209
40495669
11230
40503798
11210
40496056
11231
40504184
11211
40496444
&«.
&c.
Here it is seen that the leading figures 049 are those, not only
of the mantissa of the logarithm of 1 1200, but also of all the
following ones that come before the mark *,
This mark ♦ which stands for 0, is used to give notice that for
the mantissa in which it occurs, and for all succeeding mantissas
to the end of the line, the three leading figures are to be found in
the next lower line and under an asterisk standing opposite the
character ♦ ; whereas in other cases, the three leading figures of
any mantissa are in the same line with the other four, or are the
first to be found in a preceding line.
The use of the asterisk in connexion with the sign ♦ , is to
mark more distinctly the line in which a change takes place in
the three leading figures of the mantissa.
20. The difference of the first two mantissas on the page be-
fore us, which are .0492180 and .Q492568, is if we neglect the
decimal point, 388 ; the next difference viz. that of 0492568 and
0492956, is also 388 ; the next is 387 ; and so on in the following
series, 388, 388, 387, 388, 387, 388, 387, 388, 387 ; these differen-
ces being found smaller and smaller as we proceed to the b6ttom of
the page, where the difference between two consecutive mantissas
c
(18) INTRODUCTIOlf.
is 368. Now the differences 388, 387, 386, 385, &c. are placed
in the column headed Diff. Sf MuU. each opposite the number 1 ;
and the multiples of each difference by the numbers 2, 3, 4, . . . 9,
axe placed under that difference, and opposite these multipliers.
The use of the multiples and differences will be stated below.
From page 22 onward, there is a column marked Diff. which
contains simply the differences between consecutive logarithms
in the table, and is of but little use.
The manner of using the table of logarithms, may now be con-
veniently explained.
To find the logarithm of a given number.
21. If after rejecting ciphers at the beginning or end of the
number when they occur, there are not more than three remain*
ing figures, these figures are to be found in one of the columns
headed N, on one of the first three pages of the table, aiKl op-
posite the number, in the next column headed Log. will be found
a number which with the decimal point prefixed is the mantissa
of the required logarithm, carried to seven places of decimals.
The characteristic of the logarithm must be supplied according
to the rule in Art. 6.
The first three pages of the table are to be used only in the case
just specified. The remainder of the table must be employed in
every other case, and is available in that case also, though not so
convenient then as the first three pages.
If then the first three pages of the table be disregarded, the
general method of finding the logarithm of a number is the
following.
22. First suppose the given number to consist of five integral
figures not followed by a decimal.
The first four figures of the number will be found on the left
in the column headed Num. of one of the pages of the table :
and if the fifth figure is 0, the required logarithm carried to
seven places of decimals, will be found opposite the first four fig-
ures in the next column headed ; or at least the four last of the
decimal figures will be found there, and the place of the three
leading figures when vacant, must be supplied by those of the
first previous logarithm in which all the seven decimal figures
appear : but if the fifth figure of the given number is one of the
nine digits, it will be found at the top of one of the nine columns
which follow the column headed 0, and under this fifth figure
TABLE OF LO€ABITHMS. (19)
«nd opposite the first fonr will be found the last four decimal fig-
ures of the logarithm sought : the three leading figures are to be
looked for in the column headed 0, and opposite the other four ;
and if not found there, they may be found a little above in the
same column, unless the character ♦ either stands in place of one
of the four decimal figures first obtained, or is somewhere on the
left in the same line with them, in which case the three leading
figures are to be found directly under an asterisk, that stands op-
posite the mark ♦ . The jptofet characteristic viz. 4, is to be pre-
fixed to the seven decimal figures.
Thus the Ic^arithms of 67100, 10350, 57680, 57708, 42083,
44978, 23449, are 47566361, 40149403, 47610253, 47612360,
46241067, 46530001, 43701243, respectively.
23. Next if the number concerned consist of Jive integral fig-
uies with a decimal fraction annexed, find the logarithm of the
fi^e integral figures by the preceding rule, and to this add a cor-
rection for the additional decimal ; which is found by multiply-
ing the decimal into the difference between the logarithm taken
£rom the table and the following one.
Thus the logarithm of 10800.76 is found by taking the loga-
rithm of 10800 from the table, and increasing this logarithm by
the product of .76 into the difference between the logarithm of
10800 and the next one in the table, viz. that of 10801. Now
the logarithm of 10800 is 40334238, and that of 10801 is
40334640 ; their difference viz. 402 multiplied into .76 gives
305.62, which we may call 306, taking the nearest whole num-
ber. This added to 40334238 will give 40334544 for the log-
arithm required.
It will be observed that the difference 402 has been regarded
as a whole number, whereas it is really a decimal, the last figure
2 belonging to the sevmith decimal place. But this can lead to
no error, if after obtaining the correction 306 we make its last
figure fall in the seventh decimal place when we add it to the
logarithm 40334238.
The logarithms of 87012.9 and 27018.61 found in this way,
are respectively 49395836 and 44316630.
24 The rule given for correcting a logarithm as found for five
integral figures of a number, on account of an additional decimal,
is derived from the consideration, that whatever is the difference
between the logarithms of one number and the next in the table,
the difference between the logarithms of the second number and
(20) IHTRODDCTION.
the succeeding one is the same, or is only one less ; and so on :
for, if for the difference of a unit in two numbers there is a cer-
tain difference in their logarithms, and for a difference of two in
the numbers, twice the former difference in the logarithms, and
so on, then we may infer that for half a unit added to the least
number, there should be added to the logarithm of that number
half the difference between this logarithm and the next, and in
short, for any decimal part of a unit added to the least number,
there should be added to its logarithm a proportional part of the
difference between this and the next logarithm.
For example, the logarithm of 27018 is 4.4316532, that of
27019 is 4.4316693 which exceeds the former by 161, and that
of 27020 is 4.4316853, being 160 greater than the preceding and
321 greater than the first logarithm : if then with an addition of
OTie to the number 27018 there is an addition of 161 to its loga-
rithm, and with an addition of two the addition to the logarithm
is very nearly twice as great, an addition of .61 to the number
should give an addition of .61 X 161 to its logarithm.
25. The logarithmic correction for the decimal part of a num-
ber may be considered as made up of corrections for the several
figures of the decimal ; and instead of multiplying the whole
decimal at once into the difference of the logarithms concerned,
we may multiply this difference by the first figure of the decimal,
and as this figure stands for tenths, divide the product by 10,
which will give the correction due to the first decimal figure ;
then multiply the difference by the second figure and divide by
100, to obtain the correction for that figure ; next multiply by
the third figure and divide by 1000, to obtain the correction for
the third figure, and so on ; and lastly take the sum of all these
partial corrections, for the total correction required.
But the labor of such multifrfications may be saved by using
the columns of differences and multiples given in the table ; for
among these differences we may find the difference between any
two consecutive logarithms of the table, and under it, its multi-
ples by all the numbers 2, 3, 4, 6, 6, 7, 8, 9.
26. To obtain by this means the logarithmic correction due
to the decimal part of a number, find the difference between the
logarithm to be corrected and the next, look for that difference
under the heading LHff. Sf MuU. and from it together with its
eight multiples as found there, select the numbers corresponding
TABLE OF LOGARITHMS. (21)
to the different figures of the decimal ; then having divided the
number which belongs to the first decimal figure by 10, that
which belongs to the second by 100, and so on, add together
the quotients ; their sum will be the required correction. When
one of the figures in the decimal concerned is 0, there will of
course be no partial correction due to it.
To find for example, the logarithm of 98741.207.
The logarithm of the whole number 98741 is 49944976 ; the
difference between this and the next tabular logarithm is 44 ;
under 44 in the table of differences and multiples and opposite 2
is the number 88, which divided by 10 gives 8.8 for the first par-
tial correction ,- the second figure in the decimal .207 being
may be disregarded ; for the third figure the correction is .308,
as 308 is the multiple opposite 7. The whole correction then is
8.8+. 308=9.108 or 9 nearly; and the required logarithm is
4.9944984.
It is more convenient and sufficiently accurate to take accoimt
of one decimal figure only, in each partial correction. Thus
in place of .308 in the preceding example, may be substituted
simply .3.
27. In ascertaining the difference between one logarithm in
the table and the next, it will generally be sufficient to find the
last figure of this difference ; for this will show which of the dif-
ferences at hand under the heading Diff. 4* MuU. is the one to
be employed, as two that end with the same figure are not to be
found near each other any where in the table, except on the 4th
page. Thus in the case of the example preceding, the difference
between 9944975 and 9945019 is obtained by taking 6 from 9,
which shows that the difference ends with 4, and then looking in
the table of differences and multiples for the nearest difference
ending thus, which is found to be 44.
28. If the number whose logarithm is sought YiBsfour or more
decimal figures, the logarithmic correction due to all after the
third will be extremely small, and the required correction may
be obtained with sufficient accuracy from the first three figures ;
the third figure ought however to be increased by one^ when
the first neglected figure is 6 or more. Thus instead of the num-
bers 27109.346602, 10340. 18%01, we may employ 27109.346,
10340.19.
29. We have hitherto supposed the number whose logarithm
is sought, to have five integral figures with or without a decimal
(22) INTRODUCTION.
annexed ; but if this is not the case, we may proceed to find its
logarithm as if the first five of its significant figures were integral
and the rest decimal, only taking care to substitute instead of the
characteristic 4, the true characteristic of the logarithm of the
given number as determined by the rule in Art. 6. For the loga-
rithms of numbers having the same significant figures differ only
with respect to the characteristic, as has been shown in Art 15.
If the number proposed has less than five significant figures,
the deficiency may be made up by the addition of ciphers.
30. The use of the table will now be illustrated by a few ad-
ditional examines.
To find the logarithm of 870358.63.
Logarithm of 87035 without the characteristic, .9396939
Addition for first figure of .863 .... 40
" second " . . . . 3
" third " .... 0.2
Logarithm of 870358.63 with proper characteristic, 5.9396982
To find the logarithm of 0.14031897.
Logarithm of 14031 1470886
Correction for first figure of .897 . . . 248
" second " .... 27.9
" third « ... 2.2
Logarithm of .14031897 .... 1.1471164
To find the logarithm of 20.2147839.
Logarithm of 20214 3056523
Addition for first figure of .784 .... 149.8
" second " .... 17.1
" third " .... .9
Logarithm of 20.2147839 .... 1.3056691
To find the number answering to a given logarithm.
31. First suppose the characteristic of the l(^;arithm to be 4;
the number sought will then have five integral figures. The
first three figures of the given mantissa are to be found in a col-
umn headed 0, and the other four are to be looked for in the
same column or in one of the nine following columns ; and if
they are there, the number sought will consist only of five in-
tegral figures, the last of which will be found over the four final
TABLE or LOGARITHMS. (23)
figures of the mantissa, at the head of the column, and the other
four will be found opposite the same final figures, in the colunm
on the left headed Num. ; supposing the vacant places of that
column to be filled with their appropriate figures
Thus the number answering to the logarithm 43321758 is
obtained by turning to a part of the table where the leading
decimal figures of the logarithm are 332, as in page 23 ,* we find
there a mantissa exactly equal to the giren one viz. 3321758, and
supplying 4 the omitted characteristic of the table, we have
the given logarithm 4.3321758 ; the number answering to which
is 21487.
In like manner the numbers answering to the logarithms
4.5274108, 48430022, 46270883, 47869643, wUl be found to
be 33683, 69663, 33668, 61230.
But if the given logarithm is not to be found in the table,
falling between two consecutive logarithms found there, then the
five integral figures of the number answering to the least of these
l(^arithms, taken from the table as in the preceding case, will
form the integral part of the required number ; to which is to be
annexed a decimal correction, for the excess of the given loga-
rithm above the tabular one that answers to the five integral fig-
ures. The correction is found by dividing the difference of the
given logarithm and the tabular logarithm next below it, by the
difference between the two tabular logarithms next above and
next below it.
This correction depends upon a comparison of the given loga-
rithm with the two in the tables between which it falls, and is
derived from the following proportion, which though not strictly
accurate is nearly so.
The difference between the tabular logarithms, is to the differ-
ence of the corresponding numbers which is 1, as the difference
between the least of these logarithms and the given one, to the
difference of their corresponding numbers.
As the third term is less than the first, the fourth will be less
than the second, that is less than 1. This term, which is found
by dividing the third term by the first, when annexed as a deci-
mal to the five integral figures corresponding to the least of the
two tabular logarithms, will give the logarithm sought. Thus,
if we seek the number whose logarithm is 4.7875397, this loga-
rithm not being in the table, we find there two consecutive man-
tissas .7876384, .7875455, that with the addition of the omitted
(24) INTRODUCTION.
characteristic 4 form two logarithms, of which one is greater and
the other less than the given one ; the number answering to the
least of these is 61311, which is the integral part of the required
number ; to find the decimal part, we divide the difference be-
tween 397 and 384, by the difference between 455 and 384 ; the
quotient viz. 4? ^r .183 nearly, added to 61311 gives 61311.183
for the number sought.
32. In converting 4 f to a decimal fraction by the common
method, we annex a cipher to 13, and for the first decimal figure
take the nmnber which multiplied by 71 will equal 130 or fall
just below it ; then after subtracting the product from 130 and
annexing a cipher to the remainder, we proceed in the same way
to find a second decimal figure, and so on.
Now this process may be shortened, by aid of the table of dif'
ferencea and multiples : for in that we find the difference 71 to-
gether with its multiples by all the digits 2, 3, ... 9 ; having
then annexed a cipher to 13, we see by glancing at these multi-
ples, which one equals or falls next below 130 ; the digit oppo-
site this is the first decimal figure sought ; we then subtract this
multiple from 130, and after adding a cipher as before, look again
for the next lower multiple, opposite which is the second dec-
imal figure sought ; and so on.
Should any remainder, after having a cipher annexed, fall be-
low the difference 71 itself, the decimal figure sought will be 0,
and we may proceed immediately to find the next, by annexing
an additional cipher to the remainder.
33. If instead of annexing ciphers to the remainders and sub-
tracting the entire multiples, we leave the remainders unaltered
and divide the multiples by 10, 100, 1000, &c. we shall obtain the
same or very nearly the same result. It will be best, however,
not to do this till after the first decimal figure is obtained. Thus
in the preceding example, after obtaining the first decimal figure
1 and subtracting the corresponding number 71 from 130, we
may seek the multiple which divided by 1 or with the omis-
sion of its last figure, shall equal or just fall below the remainder
69 ; this we shall find to be 668, opposite which is the digit 8,
the second decimal figure sought ; 668 divided by 10 is 56.8 or
nearly 67 ; this taken from 69 leaves 2 : we are now to look
for the multiple which divided by 100 shall differ least from 2 ;
this is 213, opposite which is the number 3, the third decimal
figure sought.
TABLE OT LOGARITHMS. " (25)
In like manner let the numbers answering to the logarithms
4.8111890 and 4.1610157 be found: they are 64742.433 and
14488.243.
34 We have thus far supposed the characteristic of the given
logsffithm to be 4
But if the logarithm have a dijfferent characteristic, we must
seek the corresponding number as if the oharacteristie were 4,
and instead of so placing the decimal point that the number shall
have five integral figures, give it the position required by the true
characteristic of the logarithm according to the rule in Art. 6 :
for if two logarithms differ only in their characteristics, the num-
bers answering to them will have the same significant figures,
and will differ only as to the positions of their decimal points.
Thus to find the number whose logarithm is 3.9595809, look
for it as if the characteristic of the logarithm were 4 ; but instead
of placing the decimal point after the fifth figure, place it after
the fourth, so that the number of integral figures may be one
greater than the characteristic 3. The number so obtained will
be 9111.3124.
In like manner shall we find that the numbers answering to
the logarithms 1.6360175, 2:7583012, are .43253129, .057319343,
respectively.
35. We have then this general rule for finding the number
answering to a given logarithm.
If the mantissa of the logarithm is to be found in the table,
look opposite it in the column headed Num. on the left, for the
first four figures of the number sought, and over it for the fifth
figure. Then determine the place of the decimal point from the
characteristic of the logarithm, according to the rule in Art. 6 ;
annexing or prefixing ciphers, if necessary.
But if the mantissa of the logarithm proposed is not to be found
in the table, take the one next below it ; opposite this will be
found the first four figures of the required number, and over it
the fifth : then to find a few additional figures, (not more than
three,) take the difference between the mantissa of the given log-
arithm and the mantissa next below it in the table, and also the
difference between the latter and the next greater one in the ta-
ble ; look under the heading Z>{^. Sf Mult, for the latter differ-
ence, and firom it with its eight multiples given there, select the
number which is either equal to the former difference multiplied
(26) XNTBOOOrCTION.
by 10 or just below it ; opposite this number will be found the
sixth figure sought; subtract the number from the diffetence
thus multiplied, and look for the largest multiple which divided
by 10 gives a quotient less than the remainder or equal to it ;
opposite this multiple will be the seventh figure of the required
number ; find the difference of the quotient and remainder, and
look opposite the multiple which divided by 100 ccmes nearest
this difference, for the eighth figure.
36. A few examples are subjoined in illusti^ion of the rule.
To Jind the number whose logarithm is 6.8110210.
First five figures of number sought 64717
whose logarithm ends with 184
First additional figure fordiff. of 210 and 184 3
found opposite the multiplier 201.
Second additional figure for diff. of 260 and 201 8
opposite which is 536.
Third additional figure for diff. of 69 and 53.6 or 54 7.
Eight figures of the number then are 64717387 ;
and the number itself is 6471738.7, having seven integral figures,
because 6 is the characteristic of the logarithm.
To find the number whose logarithm is 3.4172019.
First five figures of number sought 26133
First figure of correction for diff. 126 7
opposite which is 1162.
Correction for diff. of 1162 and 1260 5
opposite which is 830.
Correction for diff. of 98 and 83 9,
Then as the given characteristic is 3, the required number is
.0026133759.
The numbers answering to the logarithms 2.7866943,
0.0294236, 2.4220180, 8.7230263, are in like manner found to
be 611.7786, 1.070098, .026426183, 628477220.
TABLK OF I.00AK1THXIC SINES, AC. (2T)
Table op Logarithmic Sines, Tangents, &c. pp. 156-277.
37. This is a table of the logarithms of sines, tangents, se-
cants, cosines, cotangents and cosecants, carried to seven places of
decimala These logarithms are called logarithmic or artificial
sines, tangents, &c. to distinguish them from the sines, tangents,
&c. themselves, which are called natural sines, tangents, d&c.
The natural sines, tangents, &x^. are some of them greater and
others less than unity. Their logarithms must accordingly have
characteristics of which some are positive and some negative.
In otAet to render these all positive, it has been found convenient
to increase each one by 10 : no error can result from this if we
always bear in mind that they have been thus altered.
The first two pages viz. 156 and 157, being of a different
character from the rest, and indeed not a necessary part of the
table, may be here left out of view, to be subsequently described.
Pot the first 90 pages, that is from page 158 to page 247, the
sines, tangents, &c. are given at intervals of 10 seconds ; for the
remainder of the table, at intervals of one minute. At the top and
bottom of each page are numbers in heavy type, denoting degrees ;
next, in an abbreviated form, are the titles of the columns of num-
bers of which the body of the page is composed ; in the right
and left hand columns are numbers denoting minutes and tens of
seconds for the first 90 pages, and minutes only for the remain*
ing pages : in the former case, the minutes are printed in heavier
type and with the mark ', to distinguish them from the numbers
10, 20, 30, 40, 50 denoting tens of seconds.
To find the sine^ tangent^ !fc. of a given arc,
38. The following is the rule for finding the sine, tangent,
Ac. of a given number of degrees and minutes^ from 0° to 180*^.
Look for the degrees at the top or bottom of the page, and in
tfie colunan at the side of the page under or over the degrees
thus foimd look for the minutes : then if the degrees are at the
top, look for the name of the required number at the top, other-
wise at the bottom of the page ; in the column under or over
this name, and opposite the minutes already found, will be the
number sought.
Thus to find from the table the sine of 18^ 20^, we first look for
18^ which we find at the top of the 25lst page, and in the col-
(26) INTBODUCTION.
umn under this at the side of tiie page we look for W : we then
look for the name of the required number viz. sine, at the top of
the page, because 18^ is there ; and in the column under the
name sine and opposite the 30 minutes, we find the required
number which is 9.4976824 ; the omitted characteristic 9. being
supplied
The characteristic when omitted is always to be supplied as
here.
In like manner it will be found that the cotangent of 126^ 44^
is 9.8729032, the secant of 89^ 2" is 11.7728666, and the cosine
of 1750 47' is 9.9988228.
If there are tens of seconds in addition to the given number
of degrees and minutes^ and the degrees are found on one of the
first 90 pages, then by a similar method to the one just stated,
we may find the sine, tangent, ice, of this number of degrees,
minutes and seconds. We are to look for the degrees and min-
utes as before, and then finding the seconds between the given
number of minutes and the next greater number of minutes, look
opposite them for the number sought. Thus the cosine of
140 5' 40'' is 9.9867250, the omitted figures 9.986, being sup-
jdied; and the tangent of 100° 37' 20" is 10.7268913.
39. In all other cases the given number of degrees, minutes
and seconds will not be found exactly in the table ; but the given
number of degrees and minutes, and frequently the even tens of
seconds also, will be found there, with the corresponding sine,
tangent, &c. and next to this sine, tangent, &c. will be found
the sine, tangent, &c. for one minute or as the case may be for
10 seconds more ; and the required sine, tangent, &>c. will fell
between the latter and the former : to obtain it, take from the
table the former sine, tangent, &c. viz. that which belongs to
the given number of degrees and minutes, or degrees, minutes,
and even tens of seconds ; and for the additional seconds prepare
a correction, by multiplying them into the number that is found
in the nearest column headed diff. 1', opposite the place where
the number sought would fall if inserted in the table : then if
the tabular number to be corrected is the least of the two consecu-
tive ones between which the required number falls, the correction
thus obtained is to be added to it ; but if it be the greatest, the
correction is to be subtracted from it.
Thus to find the tangent of 19<=^ 27' 45".4, take firom the table
the tangent of 19^ 27' which is 9.5479430, and for the seconds
TABLE OF LOeABITHHlC SIKE8, AC. (29)
4S^'.4 obtain a correction by multiidying 45.4 into 67.03 the
number found in the neaiest column headed dif. V^ opposite the
{dace where the required tangent would fall if inserted in the
table ; for it would fall between the tangents of 19^ 27^ and 19^
28'. The correction thus obtained is 3043.162 or 3043 if the
firaction .162 be neglected. Then as the tangent of 19^ 27^ is
less than th^ of 19^ 28', the tangent sought is 9.5479430+3043
or 9.5482473, the correction being applied to the decimal part of
the tabular number, as it should always be.
40. In explanation of the preceding rule, it may be obs^rred
that as the sines, tangents, &,c. in the table, for the first 90 pages,
are given at intervals ol 10 seconds, the difference between any
two consecutive ones is a difference fur 10 seconds ; and the part
of this difference due to each of the 10 seconds would be a tenth
of it, if the differences were the same for all intervals of the same
number of seconds : though this is not strictly true to any extent,
yet if we regard it as true for an extent only of 60 seconds or less,
the resulting error in most cases will be extremely small. Ac-
cordingly, in the first 90 pages of the table, one tenth of each dif-
fidence between two consecutive nmnbers is given, with the im-
{^ed assumption that it would be the difference of consecutive
sines, tangents, &c. for intervals of V each, near the place where
it occurs.
And in the latter part of the table, where the intervals are a
mimUe each, the differences given for V* are the numbers which
result from dividing the differences of the consecutive sines,
tangents, &.c. by 60.
Having thus the variations of sines, tangents, &c. given for
intervals of 1'', we can readily find by multiplication the varia-
tions for any small number of seconds, and thus from a tabular
sine, tangent, &c, near the required one, obtain the required one
itself, by applying to the former a correction for the number of
seconds' interval between the two.
The number of seconds for which a correction is required will
always be less than 60.
41. It will be seen that each column of differences is made to
answer for the two columns of tabular numbers on each side of
it ] one for sines and cosecants, another for tangents and cotan-
gents, and a third for secants and cosines. The reason of this
will be obvious if we consider that the natural sine of an angle
multiplied by the cosecant of the same angle, equals the square
(30) INTRODUCTI^K.
of the radius or unity ; that the Ic^arithm of the sine added to
the logarithm of the coeecant accofdingly equals which is the
Logarithm of unity ; and if 10 be added to the logarithmic sine
and cosecant sev^ally, their sum will become 20. The sum of
the numbers given in the table for the sine and cosecant of the
same angle, should then be 20 in every case ; and if in passing
through an interval of V or any number of seconds, the aine in-
creases by a certain amount, the coeecant must, for the same in-
terval of variation, decrease by the same amount. The corres-
ponding sines and cosecants therefore vary by equal diflEerences,
but in opposite ways.
The same may be said of tangents and cotangents, and also of
secants and cosines.
42. In finding a correction to be applied to a tabular number,
we may perform the required multiplication in an abbreviated way.
Let the figures of the multiplier be taken in an inverted order,
and let the unit figure be put under the place of tenths in the
multiplicand ; then if each figure of the multiplier be multiplied
into the figure above it and into all the preceding figures of the
multiplicand, the following ones being neglected, we shall have
several partial products, in each of which the last figure belongs
to the place of tenihSj and the sum of which, if we neglect the
final figure, will be the whole correction sought.
For exami^y if the difference for 1'' taken from the table is
67.91, and the number o( seconds for which the correction is to
be made, 17^^32, the method of procedure is as follows.
67.91
23 71
679.1
475.3
20.1
1.2
1176.7
The correction therefore if we take the nearest whole num-
ber is 1176.
Instead of whdly neglecting portions of the multiidicand in
the manner specified above, it will be better to regard them so
iar as is necessary to see what number must be carried from each
to make the right hand figure of every partial product as correct
as may be.
TABLE or LOOABITHMIC 8IIT£8, AC. (31)
V
This method of multytieation nuy be applied to the preceding
example thus,
67.91
23 71
679.1
476.4
20.4
1.4
1176.3
The integral result viz. 1176, does not differ firom that before
obtained : but in some cases one or two units would be gained
in accuracy by the change in question.
43. Two or three examples may now be given, in further il-
lustration of what may be deemed the best method of using the
table of sines, tangents, &c.
To find the cosine of 123° 11' 26".7.
cos. 123° ll'=9.7382412,
diff. l''»32.18, to be multiplied as below by 26.7.
32.18
7 62
643.6
193.1
22.6
869.2
The resulting correction 869, added to the above tabular number,
gives 9.7383271 for the required cosine.
To find the cosecant of 79° 14' 17".3.
cosec. 79° 14' 10"= 10.0077094,
diff. 1"=40, which multiplied by 7.3
gives the correction 29. This taken from the tabular number to
be conected, leaves 10.0077066 for the cosecant sought
To find the tangent of 168° 36' 47".8.
tang. 168° 36' 40"=9.3041318,
diff 1"= 108.7,
108.7
87
760.9
87.0
(32) INT&0]>UCTl<rH.
Subtracting the conection 848, we obtain 9.3040470 for the tan*
gent sought.
To find the arc anstoering to a given sine^ tangent^ 4^.
44. As the sines, tangents, &c. given in the table are those of
an exact number of degrees and minutes, or degrees, minutes
and tens of seconds, if the proposed number happens to be die
same with one of those in the table, it will of course be the sine^
tangent, &c. of an exact number of degrees and minutes, or of
degrees, minutes and tens of seconds.
Then look for the given sine, tangent, dtc. in a column of the
table over or under the appropriate name ; and if it is found there,
the number of degrees required will be found at the top or bottom
of the page, according as the name of the given number is at
the top or bottom of the column in which it is found; and
under or over the number of degrees, in the column at the
side, the required number of minutes or as the case may be of
minutes and tens of seconds, will be found opposite the given
number. Thus the sine 9.9651953 will be found to be the sine
of 67^ 2Z\ For, looking in the columns marked gin. at the
top or bottom, we find the proposed number on page 255, in a
column having the title ein, at the bottom ; we therefore take
the number of d^rees given at the bottom of the page viz. 67^,
and over this number of degrees, in the right-hand column, we
find 22 standing opposite the given sine, for the required number
of minutes.
In like manner shall we find that the tangent 9.3178527 is
the tangent of 11^ 44' 40"', the secant 10.8585473 is the secant
of 82^ 2' 20^'.
45. There are two numbers of degrees at the top and also at
the bottom of each page, one less and the other not less than 90^.
Their sum is invariably 179° ; and under or over each is a col-
umn of minutes or of minutes and tens of seconds, at the side
of the page, so arranged that the sum of every two opposite num-
bers in the two columns is GO' or 1°. This added to the for-
mer sum makes 180° ; so that the numbw of degrees and min-
utes, as found on the right, is in every case the supjdement of
the corresponding number of degrees and minutes found on the
left hand.
TABLE OF LOGAaiTHMIC SINES, &C. (33)
Either of these numbers belongs as properly to the sine, tan-
gent, &c. corresponding to it in the table, as the other ; for the
sine of an angle and that of its supplement are the same : so
is it with the cosecant Also the cosine of an angle and that
of its supplement agree in numerical value, though they have
opposite signs. The same is true of the secant^ tangent and
cotangent.
But in practice, the nature of each case will generally deter-
mine whether the required number of degrees is or is not less
than 90O.
46. If the proposed sine, tangent, &c. is not to be found ex-
actly in the table, there are however two successive ones between
which it falls ; then look for these, and taking that which is op-
posite the least number of minutes or tens of seconds, find th^
corresponding number of degrees and minutes, or degrees, min-
utes and tens of seconds, as in the preceding case ; and to correct
this for the difference between the tabular sine, tangent, &c. just
spoken of and the given one, add such a number of seconds as
shall result from dividing that difference by the difference for V
as found at the proper place in the table. The sum will be the
required number of degrees, minutes and seconds.
Thus, to find the number of degrees, minutes and seconds
answering to the sine 9.9766321.
On the 251st page, in the column marked sin. at the bottom,
will be found two consecutive numbers between which the pro-
posed sine falls, one opposite 22 the other opposite 23 minutes ;
taking the former, we find 71° 22' for the corresponding number
of degrees and minutes : the difference between the given sine
and 9.9766171 is 150 ; this divided by 7.10 the difference for 1''
as found in the table, gives 21'M3 for the number of seconds
which must be added to 71° 22^ to make up the required num-
ber of degrees, minutes and seconds.
In like manner the cosine 9.8055233 will be found to be the
cosine of 50° 16' 48''.88 ; the tangent 9.3768555, the tangent of
13° 23' 44".21; the cosecant 10.0059330, the cosecant of
80° 33' 03". 14
In each of these cases there is also a number of degrees, min-
utes and seconds greater than 90°, answering to the proposed
sine, tangent, &c.
£
(34) IHTBODUCTIOlf.
47. The number of seconds to be obtained by division accord*
ing to the preceding rule, may be found most conveniently by
cutting o£f figures from the divisor, instead of annexing ciphers
to the dividend when there would be occasion for this in the
common mode of dividing. It will be best however to add one
cipher to the dividend at the outset. And when the dividend
is thus multiplied by 10, if we multiply the divisor also by 10,
the value of the quotient will not be affected.
Now for the first 90 pages of the table, the divisors (which are
the tabular differences for 1'^) have one decimal figure, and mul-
tiplying them by 10 makes them all integral. So far as these 90
pages are concerned, the following example will serve to show
how the division is to be performed.
Let 2442 be the number to be divided, and 335.3 the tabular
difference for V^ by which it is to be divided. Then annexing a
cipher to 2442 and dividing by the whole number 3353, we ob-
tain 7 for the first figure of the quotient and 949 for a remainder ,*
if the division is continued, the rest of the quotient will be a
decimal : to obtain the next figure, instead of annexing a cipher
to 949, we cut off the right hand figure of the divisor and divide
949 by 335 ; the quotient is 2 : this multiplied by 335 gives 670,
and if we have regard to the figure cut off, so far as to see what
ought to be carried from it to the product obtained from the other
figures, we find that this product must be increased by 1, as six
tenths is nearer one than nothing; then subtracting 671 from
949 we find the remainder 278 which divided by 33 gives 8 for
the third quotient figure ; multiplying by this and carrying 4 as
we should do, we obtain 268 ; this taken from 278 leaves 10,
which divided by 3 gives 3 for a fourth quotient figure. This
process may be exhibited as follows.
3353)24420(7.283
23471
949
671
"278
268
"lo
10
It will hardly ever be worth while to seek for more than ttvo
decimal figures.
TABLE or LOOABITHMIC SINES, d&C. (30)
48. From page 248 onwards, the tabular differences for V are
carried to two places of decimals. If these when used for divi-
sors be regarded as integral after the dividend is multiplied by
10, the quotient will be ten times too small ; but it may readily
be made correct by changing the place of the decimal point.
For example, to find the quotient of 1182 2148)11820(5503
divided by 21.48, we may divide 11820 by 10740
2148, as in the scheme annexed, which
will give us the figures of the required quo- 1080
tient ; but in order that they may have their 1074
local values, the decimal point must be put
after the second figure instead of the first. 6
The quotient as thus determined is 55.03. 6
49. The following examples may be of use in illustrating
what precedes.
Required the number of degrees^ minuter and seconds, whose
mW is 9.7035391.
The number 9.7035329 is the sine of 30^ 21' ; the diflerence
between 29 and 91 is 62 ; the tabular difference for V is 35.95.
Then, 3595)620(1.72
360
260
252
~8
7
The answer therefore is 30^ 21' 1''.72.
Required the number of degrees, minuses and seconds, whose
cotangent is 10.8683783.
10.8685308 is the cotangent of 7° 42' 30" ; the difference of
5308 and 3783 is 1525 ; the tabular difference for 1", 158.4
Then,
1584)15250(9.63
14256
994
950
48
The answer therefore is 7° 42' 39".63,
(36) INTRODUCTION.
We may now explain the use of pages 166 and 157.
60. By turning to page 168 and several of the following pages, it
will be seen that the numbers in the columns marked diff.'l'' vary
rapidly, the difference for the first interval of W being infinite,
for the next 301030, for the next 176091, &c. whereas these
differences further on in the table vary slowly, being nearly the
same for several successive intervals of 10''. Now in cases where
the sine, tangent, &rC. are not found in the table, we cannot deter-
mine them accurately by the rules we have given for the purpose,
unless the differences in question vary slowly ; as those rules are
founded on the supposition that the differences are, for short dis-
tances in the table, nearly uniform. Accordingly, the 166th and
167th pages are prepared as a substitute for the twelve subsequent
pages, to be used when strict accuracy is required.
In the columns headed sin. tan. cot. cosec. are given the loga-
rithmic sines, tangents, cotangents and cosecants, to every minute
of two degrees : the columns headed sec. and cos. contain the
corresponding secants and cosines ; of these however, all the
figures except the last four are in each case to be found at the
head or at the foot of the column. The left hand column of
page 166 presents the number of seconds in each number of min-
utes opposite it in the next column. And in the left hand co-
lumn of page 157, is given the number of seconds for every num-
ber of minutes from 1^ or 60' to 2^ or 120'.
Under the heading 5^ - n are two columns of numbers, each
consisting of four figures that are to be preceded by the figures
4685 which stand above and below these columns. Each num-
ber in the first of these two columns is the excess of a logarithmic
sine opposite and adjacent to it, above the logarithm of the oppo-
site number of seconds at the side of the page on the left ; and
each number in the other column is the excess of the opposite and
adjacent logarithmic tangent, above the logarithm of the opposite
number of seconds. Thus 4.6855747 is the excess of 7.1626960
over the logarithm of 300 which is 2.4771213 ; and 46855788
is the excess of 7.7190026 over the logarithm of 1080 which is
3.0334238. The numbers in the columns imder the heading
q+n, sie such as result from adding to each cotangent and cose-
cant in the adjacent columns, the logarithm of the opposite num-
ber of seconds. Thus 16.3144197 is the sum of 12.2140492 and
TABLE OF LOGARITHMIC SINES, &C. (37)
the logarithm of 1260 which is 3.1003705; and 15.3144398 is
the sum of 11.8460925 and the logarithm of 2940 which is
3.4683473.
51. If the sines in the column headed sin. were not given, they
might be found from the numbers in the next column. For if q
be put for the sine of any number of minutes less than 120^ or 2^
and n for the logarithm of the corresponding number of seconds,
the value of q-n will be given by the table ; which value in-
creased by n will be the value of q the required sine. If for
example the sine of 1^ 23^ be required, we shall find by looking
in the first of the columns headed q^n on page 157, that
y- 71=4.6855327: therefore 5r=46855327+n; but the loga-
rithm of 4980 the number of seconds in 1° 23' is 3.6972293 :
hence 5^=8.3827620, the sine required.
In a corresponding manner may tangents, cotangents and cose-
cants be found. To find for example the cosecant of 0° 37',
we obtain from the table the equation gr+n =15.3144335, whence
9= 15.3144335 - n ; and as the number of seconds in 37' is 2220
whose logarithm is 3.3463530, ?= 11.9680805.
52. The numbers in the columns headed q-^n and q+n, are
given only at intervals of V or 60" ; but they vary so slowly and
regularly, that we may by a simple proportion find the correspond-
ing numbers for intervening points with facility and accuracy.
To find for instance in the first of these columns the number
belonging to (P 40' 37", we observe that its last four figures must
be less than 5651 but greater than 5646 : the difference of these
two numbers viz. 5 being a variation for an interval of 1' or 60",
the part of this variation for 37" may be found by the proportion
60 : 37 : : 5 : a:= 'jV =3 nearly. The last four figures of the re-
quired number are therefore 5648; and the number itself is
4.6855648. In like manner may we find the number that be-
longs, in either of the four columns headed q-n and y+n, to
any number of degrees, minutes and seconds from 0° to 2^.
53. If the numbers thus found by interpolation be increased or
as may be necessary diminished in each case by the logarithm of
the corresponding number of seconds, the corresponding sines, tan-
gents, cotangents and cosecants will be obtained. Let it be required
for example to find the sine of 1° 3' 41". 79. By inspecting the
column headed sin. on page 157, we shall find that the required
sine is between 8.2630424 and 8.2698810. But to obtain it ac-
(38) INTRODUCTION.
curately by interpolation in this column would be inconrenient,
as the differences of the numbers there are large and vary rapidly.
If however, we put q for the required sine and n for the loga-
rithm of the corresponding number of seconds, we can easily
obtain the value of y - n from the proper colunm headed q — n:
this value is 46855500 ; the correction for 41^'.79 being nearly
6, which has to be taken from 5506. Now the number of
seconds opposite P 3^ in the table is 3780, which increased by
41.79 becomes 3821.79 whose logarithm we shall find to be
3.5822669; this is the value of n, and since f -n=46855500,
9 = 8.2678169, which is the sine required.
Again, to find the cotangent of 0° 57' 34'^46.
9+n== 15.3143845
n=: 3.5383802
.*. 9 =11.7760043, the cotangent sought.
54. In like manner may these two pages be used, when the
number of degrees, instead of being 0^ or 1^, is 88^, 89°, 90°,
91°, 178° or 179°. But if the number of degrees be found at
the bottom of the page, the title of the quantity sought must be
looked for there also ; and if the number of degrees is on the right
hsuid, the number of minutes must be found in the right hand
column ; and in this case, the opposite number of seconds in the
left hand column diminished by the given number of seconds^ is
that whose logarithm n is intended to represent.
Let it be required to find the tangent of 88° 21' 37^5.
As 88° is at the bottom of the page and on the right hand, we
are to look for the title tan, at the bottom, and for 21 the number
of minutes, in the right hand column ; opposite this number in the
first column on the other side of the page will be found 5940, for
the number of seconds to be diminished by 37.5 the given number
of seconds ; the logarithm of the remainder 5902,5 is 3.7710360,
which is the value of n. Opposite 21' and 22', and over the title
9+n, in the column next to the column of tangents, are to be
found the numbers 3051 and 3075 ; the proper intermediate num-
ber for 2V37'\5 is 3066. Therefore j+n = 15.3143066; and
hence } = 11.5432706, the tangent sought.
Again, to find the cosine of 91° 27' 43".7.
J -n=: 46855277
log. 5263.7 = 71=3.7212911
•'• q 3£ 8. 4068 188, the required cosine.
TABLE 'of logarithmic SINES, ftC. (39)
To find the cosecant of 179° ly 62'^26.
j+«=16.3144360
log. 2407.76 = n= 3.3816114
.'. q =11.9328236, the cosecant sought
55. The columns having the titles sec. cos. at the top, and
cosec. sin, at the bottom, may be used for finding sines, cosines,
secants and cosecants, in the ordinary way.
To obtain for example the cosine of 1^ 17' 26''. 6.
In the column headed cos. and opposite the 17' which stands
under 1^, will be found the number 8911 ; and opposite 18', the
number 8882 ; the proper intermediate number for 17' 26".6 is
8899, in which we have the last four figures of the required co-
sine ; the preceding figures are 9.999, as found at the top or bot-
tom of the column concerned. The complete cosine therefore is
9.9998899. Thus also the cosecant of 90° 66' 47".6 may be
found to be 10.0000672.
Instead of the columns here referred to however, it is best to
use the corresponding ones, in the twelve subsequent pages.
66. The 166th and 167th pages are likewise to be used, when
with a given sine, tangent, &c. that falls within the limits of
these pages, it is required to determine accurately the number
of degrees, minutes and seconds to which it belongs. In em-
ploying them for this purpose, we have first to find the two con-
secutive sines, tangents, d&c. between which the given one falls,
and the corresponding values of q-^n ox q+n. We may then
determine by interpolation the value of^-n or q-^-n as the
case may be, for the given sine, tangent, &c. itself; which last
quantity is represented by 7. If ^ be diminished by q-n ox
taken from q+n, the remainder will be n the logarithm of a
number of seconds, firom which the number itself is to be found,
and thence the required number of degrees, minutes and seconds.
Let it be required to find the degrees, minutes and seconds be-
longing to the tangent 8.2743291. This falls between the tan-
gents of 1^ 4' and 1° 6' opposite which in the preceding column
nnder the heading q — n are the numbers 6260 and 6266; the
intermediate number corresponding to the given tangent, will be
found by oifClinary interpolation to be 6260 ; the complete value
oi q-n is accordingly 4.6866260. Taking this from q that is
8.2743291, we find for the value of n, 3.6887031 which is the
(40) INTRODUCTION.
logarithm of the number 3878.851. The number next less than
this in the column of seconds is 3840, to which corresponds
1^ 4/. The required number of degrees, minutes and seconds is
therefore 1° 4' 38^851.
Or if we take the number of degrees found on the right hand,
we must look for the number of minutes in the right hand col-
umn, opposite the number of seconds next greater than 3878.851,
viz. 3900, and take the difference of 3900 and 3878.851 for the
number of the required seconds. In this way we shall obtain
178*^ 5& 21'M49 for the number sought ; which is evidently the
supplement of the one before obtained.
If the title of the given quantity be at the bottom of the page,
the required number of degrees is to be found there also.
57. To obtain the value o( q - n ox q+n corresponding to the
given sine^ tangent^ &c. we may find approximately by the gen-
eral table the required number of degrees, minutes and seconds,
and take the value ofy— nory+nas the case may be, corres-
ponding to the number thus found.
The required number of degrees, minutes and seconds in the
preceding example, we shall readily find on page 164 to be
nearly 1° 4' 40^^ Then as the numbers in the proper column
headed q — n on page 157 which correspond to 1*^ 4' and 1^ B'
are 6250 and 6266, the intermediate number corresponding to
\o j^t 49// is easily seen to be 6261 nearly, which differs only a
unit in the last place of figures from the result before obtained
viz. 6260. And if we take 4.6856261 for the value of ? — n,
the value of n will be 3.5887030; which is the logarithm of
3878.85. The required number of degrees, minutes and seconds
thus obtained is 1^ 4^ 38^^85, differing only a thousandth of a
second from the previous result.
58. If the given quantity belongs to either of the columns
headed sec. and cos. the required number of degrees, minutes and
seconds is to be found in the ordinary way. In this case how-
ever, it is better to make use of one of the twelve subsequent
pages.
TABLE OF NATURAL SINKS AND COSINES. (41)
Table of Natural Sines and Cosines : pp. 279-287.
69. This is a table of natural sines and cosines for every min-
ute from 0° to 9(P, carried to five places of decimals. The dec-
imal point, omitted for convenience, is to be prefixed to each sine
and cosine as given in the table.
To find the sine or cosine of a given number of degrees and
minutes.
If the given number is less than 45^, look for the degrees at
the top of the page and for the minutes on the left : then oppo-
site the minutes and under the degrees in the column headed sin.
or cos. as the case may be, will be found the required sine or
cosine.
If the number of degrees and minutes falls between 45^ and
90^, the degrees must be looked for at the bottom of the page,
and the^ minutes on tUe right : then opposite the minutes and
over the degrees in the column having the proper title at the foot^
will be found the required sine or cosine.
The sine of 7^ 33' is .13139, and the cosine of 67^ 31' is
.38241.
60. At the top of each page and on the left hand are given the
differences between consecutive numbers in the columns headed
sin. These differences are for intervals of 1' or 60'' each. In
columns under them are given the proportional parts for every
number of seconds less than 60 ; the part belonging to any num-
ber of seconds, standing opposite this number as found in the
adjacent column of minutes.
There are like columns of differences and proportional parts
on the right, over the title Diff. to be used in connexion with the
columns headed cos.
To find by aid of these columns the sine or cosine of a given
number of degrees, minutes and seconds.
Look for the two consecutive sines or cosines between which
the required one must fall ; find their difference and the propor-
tional part of it belonging to the given number of seconds ; this
added to the sine of the given number of degrees and mdnuies
or subtracted from the cosine as the case may be, will give the
required sine or cosine.
F
(42) INTRODUGTIOK.
To find the sine of 28^ 43' 5V\
This must fall between the sines of 28^ 43' and 28° 44^
The sine of 28° 43' as found in the table is 48048, and differs
26 from that of 28° 44' : 25 being a variation for 1' or 60", to
find the part belonging to 51" look in the column of propor-
tional parts under 25, and opposite 51 in the adjacent column of
minutes, where will be found the number 21 ; which added to
the sine of 28° 43' gives .48069 for the required sine.
To find the cosine of 67° 21' 37".
The cosine of 67° 21' is .38510, and differs 27 from that of
67° 22' : the proportional part of 27 for 37" is 17 ; which stUn
traded from .38510 gives .38493 for the cosine required.
61. If the number of degi-ees, minutes and seconds is between
90° and 180°, take the cosine or sine of its excess above 90°,
according as the sine or cosine of the given number is required.
Thus for the sine of 143° 20' 17" take the cosine of 53° 20' 17",
and for the cosine of 117° 26' 35" take the sine of 27° 26' 35".
If regard is to be had to positive and negative signs, the cosine
of more than 90° must be taken negatively,
62. To find the number of degrees^ minutes and seconds an-
swering to a given sine or cosine.
If the given number is the same with one of those in the table,
the required number will consist of degrees and minutes only.
Then if the column which contains the given number has the
nams of that number at the top, the required degrees will be
found there also, and the minutes will be found opposite the
number, in the column of minutes on the left : but if the name
of the given number appears at the foot of the column concerned,
.the degrees will be found there, and the minutes on the right.
The sine .92870 is that of 68° 14', and the cosine .87321 is
that of 29° 10'.
When the given sine or cosine is not to be found in the table,
look for the two consecutive sines or cosines between which
it falls ; one of these must answer to the number of degrees
and minutes required ; find the difference between it and the
given sine or cosine, also the difference between it and the other
tabular sine or cosine ; then under or over the latter difference in
a column of proportional parts, find the former difference ; oppo-
site that, in the adjacent column of minutes, will be found the
required number of seconds.
TABLE or LOGARITHMS. (43)
To find the number of degrees, minutes and seconds answer-
ing to the cosine .93208.
This falls between the cosines of 21^ 14' and 21° 15' given in
the table. 21° 14' is therefore the required number of degrees
and minutes. The cosine of 21° 14' differs 3 from the given
one, and differs 10 from the cosine of 21° 15'. On page 282 in
the column of proportional parts over the difference 10, will be
found the number 3 opposite the numbers from 15 to 20 in the
adjacent coliunn of minutes : one of these numbers is to be
taken for the required number of seconds ; let this be 18, as being
near the middle of them ; then 21° 14' 18" is the number of
degrees, minutes and seconds required.
To find the number of degrees, minutes and seconds whose
sine is .82036.
This falls between the sines of 56^ 7' and 55^ 8'. The sine
of 55^ 7' differs 4 from the given one, and differs 16 from that of
55^ 8'. The number 4 in the column of proportional parts over
the difference 16, stands opposite 15 in the adjacent column of
minutes: this is to be taken for the required number of se-
conds. 55^ 7' 15" is then the number of degrees, minutes and
seconds sought.
Table of Logarithms of Numbers : pp. 289-293.
63- This is a table of logarithms of numbers carried to five
{daces of decimals. The logarithms are given without any char-
acteristics ; and to be fitted for use, must in each case be prece-
ded by a decimal point and the proper characteristic as deter-
mined by the rule in Art. 6.
To find by this table the logarilhm of a given number,
64 If the number consist of three significant figures, look for
them in a column headed N : then in the next column headed
Ltog. and opposite those figures will be found the mantissa of
the logarithm sought
The mantissa of the logarithm of 37.9 is .57864, and the whole
logarithm 1.67864
If the number has only two significant figures, annex a cipher,
or if it has only one, annex two ciphers, and then proceed as
before.
(44) INTRODUCTION.
65. If the number consists of more than three significant fig*
uies, look for the logarithm answering to the first three signifi-
cant figures, as in the j»receding case : and then if these figures
are found on one of the first three pages of the table, look oppo-
site them in the column of proportional parts under the fourth
figure, for a correction which is to be added to the logarithm of
the first three figures ; and if there is a Jifth figure, then in the
column under this and opposite the first three figures may be
found a number which divided by 10 will give the proper addi-
tional correction : the correction due to a sixth figure may be
found in a similar manner, the divisor 10 being however ex-
changed for 100.
To find the logarithm of 30.1745.
The decimal part of the logarithm of 301 is 47857 ; the cor-
rection for 7 is 101, for 4, 5.7, for 5, .72 : the whole correction
therefore is 107 nearly. Then 47964 preceded by the decimal
point and the proper characteristic viz. 1 is the logarithm sought.
But if the first three figures of the given number with the cor-
responding logarithm are found in one of the last two pages of
the table, look on the right, in the nearest column headed D. for
the difference between this and the following logarithm; and
look for the same difference on page 293 under the heading
D. in the table of differences and proportional parts : then opposite
this difference, and under the fourth figure of the number will be
found the correction due to that figure, and under the Jifth, 10
times the correction due to that, and under the sixthj 100 times
the correction due to that ; whence may be easily obtained the
whole correction which added to the logarithm of the first three
figures of the proposed number will give the logarithm sought.
To find the logarithm of 7.41809.
The logarithm of 741 as given in the table is 86982; the
corresponding difference in the column headed Z>. is 58 ; opposite
this difference and under 8, in the table of differences and propor-
tional parts on page 293, stands the number 46, which is the
first correction ; and under 9 stands the number 52, one hun-
dredth of which is the next correction : the whole correction
therefore is 46.52 or 47 if the nearest integer be taken, and the
required logarithm is .87029.
In seeking the number which corresponds to a given loga-
rithm, the next table should be used.
TABLE OP ANTIL0GARITHM8. (45)
Table of ANTiLOGABXTHif s : pp. 294-299.
66. This is a table of numbers answering to given logarithms.
In the columns headed h, are the mantissas of logarithms carried
to three places of decimals. In each column headed Hum, are
the first jkoe figures of the numbers belonging to the mantissaa in
the jveceding column.
To fend by this table the number answering to a given logon
rithm.
67. If the mantissa of the logarithm consist of only three decimal
figures, look for these figures in the column headed L. then in
the next column headed Num. and opposite those figures, will
be found the first five figures of the required number. The
characteristic of the given logarithm will show where to place
the decimal point in this number.
To find the number answering to the logarithm 2.057.
Opposite the decimal .057 as found in a column headed L.
stands the number 11402 in the succeeding column headed
Num, and as the index of the given logarithm is 2, the number
sought must have three mtegral figures : it is therefore 11402.
68. If the mantissa of the given logarithm consists of more
than three figures, look for the number answering to the first
three figures, as in the preceding case : and then if these figures
are found on one of the last four pages of the table, look oppo-
site them on the right in the adjacent table of proportional parts
and in the column under the fourth figure, for a correction which
is to be added to the number answering to the first three figures
of the given mantissa ; and if there is a fifth figure in the man-
tissa, then in the column under this, will be found a number
which divided by 10 is the {uroper additicMial correction : if there
is dso a sixth figure, the correction due to that may be found in
the same way, except that the divisor 10 must be changed to 100.
To find the number answering to the logarithm 5.830309.
Opposite the decimal .830 in the table is the number 67608, and
under the heading 3 in the table of proportional parts is 47, the
first correction ; under 9 is 141, one hundredth of which is another
correction : the sum of the two is 48 nearly, which added to 67608
gives 67656 ; and as the index of the logarithm is 5, the number
sought must have six integral figures : it is therefcMce 676560.
(46) INTRODUCTION.
But if the first three figures of the given mantissa with the
number corresponding, are found on one of the first two pages
of the table, look on the right in the adjacent column head-
ed D. where will be found the diiference between that and the
following number, and look for the same difference on page 295
in the table of differences and proportional parts under the head-
ing Z>. then opposite this difference, and under the fourth figure
of the mantissa, will be found the correction due to that figure,
under the fifths 10 times the correction due to that, and under
the sixths 100 times the correction due to that.
To find the number belonging to the logarithm 2.016439.
The number answering to the decimal .016 in the table is 10375 ;
the adjacent number in the column marked Z>. is 24 ; opposite
the difference 24 in the table of proportional parts on page 295,
stand 10 in the column headed 4, 7 in the column headed 3, and
22 in the column headed 9 : the correction to be added to 10375
is therefore 10 +.7 +.22, or 11 nearly ; and as the index of the
given logarithm is - 2, the number sought must be .010386.
Table of Logarithmic Sines, Tangents, &g. pp. 300-32^.
69. This is a table of artificial sines, tangents, '&c. to every
minute from 0^ to 180^, carried to fioe places of decimals ; and
is designed for use in connexion with the two preceding tables.
To find by this table the sine, tangent, S(*c, of a given nuny-
ber of degrees and minutes.
Look for the number of degrees at the top or bottom of
one of the pages; and according as this number is on the
right or left, look on the right or left for the number of min-
utes : then opposite the minutes, will be found the last four or
five figures of the required sine, tangent, &c. in the column
having the corresponding title at the head or foot, according as
the degrees are found at the top or bottom of the page : the
preceding figures may be found either at the head or at the
foot of the column in which the others occur.
Thus the tangent of 96^ 36' is found to be 10.93666. In this
case the title tan. is looked for at the bottom of the page, because 96^
the given number of degrees is found there, and the minutes are
looked for in the left hand column, as 96^ is on the left of the ac-
companying number 83^.
TABLE OF LOGARITHMIC SINES, &C. (47)
When the numbers at the head and foot of a column aie not
the same, they are marked with an asteriskf and the upper or the
lower one is to be prefixed to the numbers in the intervening
column, according as these numbers are above or below that in
which occurs the character ♦ standing for 0. This number itself
must be preceded by the larger of the two numbers marked with
the asterisk.
The cotangent of 172° 64' is 10.90463,
« 1720 50^" 10.90063,
" 172° 33'" 10.88351.
To find the sine^ tangent^ Sfc. of a given number of degrees^
minutes and seconds.
70. If the number of degrees is 0°, 89^, 90^ or 179^, it will be
found at the beginning of the table on page 300 ; and the requi-
red sine, tangent, ^c. is to be found as in the case where page
166 is employed. The method of using that page is pointed out
in Arts. 61, 62, &c. One or two examples will be sufficient to
show how we are to proceed in the case before us.
To find the tangent of 90° 23' 47".
This must fall between the tangents of 90° 23' and 90° 24'.
Opposite these tangents in the adjacent column stands the num-
ber 42, which is the last pert of the quantity denoted by q+Uj
the first part viz. 16.314 being at the top and bottom of the
columns headed q+n. 16.31442 is then the whole value of q+n.
The number of seconds in the left hand column opposite 23'
is 1380 ; this added to 47 gives 1427, the logarithm of which as
obtained from page 289 is 3.16442: this is the value of n;
hence q the required tangent is 12.16000.
To find the cotangent of 89° 31' 26".
This must fall between the cotangents of 89^ 31' and 89^ 32',
opposite which in the adjacent column headed j — n are the
numbers 69 and 68, the former of which preceded by the figures
4686 over and under the column headed q—n, gives 4.68669 for
the value of ^ — n. In the column of seconds is the number 1740
opposite 31' in the right hand column of minutes ; which column
is used in this case because 89^ is a right hand number of degrees.
1740 diminished by 26 gives 1714, whose logarithm n is 3.23401.
The required quantity q is therefore 7.91960.
In hke manner the cosecant of 179° 46' 18" will be found to
be 12.39966.
(48) INTBODUCTION*
But if for example the sine of 90^ 46^ 4Z" be required, it is
sufficient to look for it as the sine of 9(P Kl\ which is found at
once to be 9.999%.
71. All of the table except the part just noticed, is to be used
according to the following directions.
Find the sine, tangent, &c. of the given degrees and minutes,
and according as this is less or is greater than the one required,
correct it for the additional seconds, by adding or by subtracting
the number in the adjacent column headed d that stands op-
posite the given number of seconds as found in the left hand
colmnn of minutes.
If however the sine, tangent, &c, of the given degrees and
minutes be found on one of the first dghi pages of the table,
correct it by adding or subtracting the product of the given num-
ber of seconds into the number in the adjacent column headed
d, 1", that stands between the tabular sine, tangent, ice, next
greater and that next less than the required one.
To find the cotangent of 72^^ 36' 42''.
The cotangent of 72° 36' is 9.49607: and in the proper colunm
headed d. is the number 31 opposite 42 in the left hand colunm
of minutes. 31 subtracted from 9.4%07 leaves 9.49576 for the
cotangent sought.
To find the cosine of 99° 27' 44".6.
The cosine of 99<^ 27' is 9.21634 ; the product of 44.6 into
1.27 the difierence for 1" as found in the table is 57 nearly,
which added to 9.21534 gives 9.21691 for the cosine required.
To find the secant of 166° 41' 38".
This must fall between the secants of 166° 41' and 166° 42'
which are 10.01184 and 10.01181, their difierence being 3: as
this is a difierence for an interval of one minute, -^^ or j\ must
be the difierence for 1" ; this multiplied by 38 is |{ or 2 neaily,
which taken from 10.01184, leaves 10.01182 for the secant
sought.
This example is designed to show how we are to proceed in
all cases when the difierences for 1" are not to be found in the
table.
To find the number of degrees, minutes and seconds belonging
to a given sine, tangent, ^c.
72. Look for the given sine, tangent, &c. in a column having
the appropriate title at the top or bottom, and if it be found there,
TABLE OF LOGARITHMIC SINES, 4cC. (49)
the required number will consist of degrees and minutes only :
then look for the degrees at the top or bottom of the page, ac-
cording as the name of the given quantity is at the top or bottom
of the column in which it is found ; and look for the minutes
on the right or left, according as the right or left hand number
of degrees is taken for the required one.
Thus if the quantity given is the secant 10.50004, we find it
on page 309, in a column having the title sec. at the bottom.
The number of degrees found there is, on the left hand 108°,
and on the right 71° ; and the number of minutes is 26' or 34',
according as it is found on the left or on the right. The given
number is therefore the secant of 108° 26' or 71° 34',
73. But if the given sine, tangent, ice. is not to be found ex-
actly in the table, look for the two consecutive ones between
which it falls, and taking that which is opposite the least num-
ber of minutes, find the corresponding number of degrees and
minutes, and for the additional number of seconds due to the
difference between the given sine, tangent, &c. and the tabular
one just specified, look in the left hand column of minutes oppo-
site this difiTerence as found in the appropriate column headed d.
Thus if the number of degrees, minutes and seconds whose
cosine is 9.49741 be sought, we readily find for the required
niunber of degrees and minutes 71° 40', whose cosine is 9.49768,
exceeding the given cosine by 27 : then in the left hand column
of minutes, and opposite 27 in the appropriate column headed d.
we find 43 for the required number of seconds. The number
sought is therefore 71° 40' 43".
If 108° be taken for the degrees instead of 71°, the minutes
will be 19' and the seconds 17" or 18".
When the required number of degrees and minutes is found
on one of the first eight pages of the table, the additional seconds
are not to be found in the manner above indicated ; but to obtain
them, we must divide the difference between the given sine, tan-
gent, dC'C. and the sine, tangent, &c. of the required number of
degrees and minutes, by the difference for 1" as foimd in the ap-
propriate column headed d, 1". Thus if the degrees, minutes
and seconds corresponding to the cotangent 10.99208 be sought,
we find 5° 48' for the required degrees and minutes, the cotan-
gent of which is 10,99321, exceeding the given cotangent by
o
(60) INTKODUCTION.
113 ; this divided by 2.10 the difference for 1" as found in the
table, gives 54 nearly for the required number of seconds.
To find the degrees, minutes and seconds whose cosecant is
10 00626.
This falls between 10.00628 and 10.00625, the cosecants of
80° 17' and 80° 18', the former of which exceeds the latter by
3 and exceeds the given cosecant by 2 : and as 3 is the differ-
ence for 1' or 60", the number of seconds to which the difference
2 is due, must be f of 60.
The number required is therefore 80° 17' 40".
This example will serve to show how we are to proceed in all
cases when the differences for 1'' are not given in the table.
74. Lastly, if the required number of degrees is found to be
either 0^, 89^, 90° or 179^, look opposite the sine, tangent, &c.
nearest the given one, for the corresponding value of 9 - n or
q+n in the adjacent column, and subtract the given quantity from
q-hn or diminish it hy q- n as the case may be, to obtain n
the logarithm of a number of seconds. Then having found the
required number of degrees and minutes, look opposite the min-
utes and in the column on the left, for a number of seconds :
the difference between this and the number whose logarithm
is n, will be the number of seconds required.
Thus if the degrees, minutes and seconds answering to the
cotangent 7.96304 be required ; we find the nearest cotangent to
be 7.96889, and the corresponding value of 9 - n to be 4.68559 ;
which taken from the given cotangent leaves 3.27745 for the
value of n. The number answering to this logarithm is 1894.3,
as may be readily found from the table of antilogarithms. The
required number of degrees and minutes is 89° 28', and 1920 is
the number of seconds on the left, opposite 28'. The required
number of seconds therefore is 25". 7, which is the difference be-
tween 1920" and 1894".3.
If however the given number belong to one of the two colmnns
headed sec. and cos. the required number is to be found directly
by inspection. Thus if the cosecant 10.00004 be the number
given, we shall find it in the table with 89° 11', 89° 12',— 89° 16',
for corresponding numbers ; either of which or rather the mean
of them viz. 89° 13' 30" may be taken for the number sought.
tbaverse table. (61)
Traverse Table, or Table of Departure akd Difference
or Latitude : pp. 323-329.
75. In this table are given the differences of latitude and the
departures, to four or five places of decimals, for distances from 1 to
9 inclusive, and for courses or bearings from 0*^ to 90^ at intervals
of 15', as also for each of the 32 quarter points of the quadrant.
These distances may be reckoned as tens, hundreds, thovr
sands, &c. 0£ tenths, hundredths, thousandths, &c. if the posi-
tion of the decimal point in each departure and difference of lati-
tude in the table be changed accordingly.
To find the departure and difference of latitude for a given dis-'
tance and bearing.
If the given bearing is less than 45^, look for it on the left in
one of the pages of the table, and look for the distance at the top
or bottom of the page : then over or under the distance and oppo-
site the bearing, will be found the difference of latitude in the
column headed diff. lai. and the departure in the next column :
but if the bearing is more than 45^, look for it on the right ; and
then for the difference of latitude, look in the column over the
title diff. lot. and look for the departure in the preceding column.
Thus for the bearing 10^ 46' and the distance 7, the difference
of latitude and the departure will be found to be 6.8772 and
1.3057, respectively.
And for the bearing 63^ 30' with the distance 30=3 tens, the
difference of latitude and the departure are 13.3859 and 26.8480.
When the given distance has more than one significant figure,
find the departure and difference of latitude for each numeral
figure of the distance having regard to its local value, and take
the sum of the several differences of latitude thus found, for the
whole difference of latitude, and the sum of the several depart-
ures for the whole departure.
Thus, if with the bearing 64^ 16' there be given the distance
63.049, we shall find the differences of latitude for the parts
of the distance represented by the several figures 6, 3, 4, 9 to be
35.055, 1.75275, .02337, .0062582: then, neglecting a few of
the last decimal figures, we may take the sum of 35.056, 1.7527,
.0234 and .0053, (which is 36.8364,) for the required difference
o£ latitude. The departure found in like manner is 51.1686.
(52) INTRODUCTION.
If the given bearing falls between two consecutive bearings in
the table, the departure and difference of latitude may be found
by interpolation in the usual way.
Thus if the distance be 4 and the bearing be 6° 20', we shall
find the difference of latitude to be 3.97553 and the departure
0.44125.
76. The traverse table will serve to show the comparative
values of the parts of any right angled triangle : for if the hypot-
enuse be regarded as a distance, and one of the acute angles as a
bearing, the opposite side will be the corresponding departure, and
the remaining side the difference of latitude.
Of the four quantities, distance, bearing, depaiture, and differ-
ence of latitude, if any two be given, tlie others corresponding to
these may be readily obtained. They are to be looked for at the
place in the table where the two given ones shall be found to
correspond.
Thus if there be given the departure 3.90518 and the diffei^
ence of latitude 0.86576, 4 will be found for the corresponding
distance, and 77^ 30' for the bearing.
If the two given quantities cannot be found corresponding
exactly in the table, the required quantities must be obtained by
interpolation.
Table of Meridional Parts : pp. 330-337.
77. This is a table giving, to every minute of latitude, the
length of Mercator's meridian expressed in geographical miles or
minutes of an arc at the equator. The miles in the length of this
meridian for any latitude, are called the meridional parts for that
latitude. These parts carried to tenths^ are placed in the table
under the degrees and opposite the minutes of the corresponding
latitudes. Thus the meridional parts for the latitude 3^ 40^ are
220.2, and for 30^ 27' they are 1919.6.
Workman's Table for correcting Middle Latitude : p. 338.
78. If the departure made by a ship in sailing on a single course
from one parallel of latitude to another, be reckoned in degrees,
minutes and seconds upon a parallel whose latitude is a mean
between the latitude left and the one arrived at, it will correspond
workman's table. (53)
nearly with the difference of longitude made by the ship. But
that the correspondence may be exact, the departure must be
reckoned upon a parallel whose latitude is somewhat greater than
the middle latitude just mentioned. The table in question is de-
signed to furnish for the most important cases, the correction that
must be added to the middle latitude to give this greater latitude.
The correction may be found in minutes, by looking under
the difference of the two given latitudes and opposite the middle
latitude.
Thus if the two latitudes concerned are i7^ and 63^, the
middle latitude is 45^, the difference of latitude 16°, and the re-
quired correction 35' ; which added to the middle latitude gives
45° 35' for the middle latitude corrected.
Tables : pp. 339, 340.
79. The use of the Table far computing Compound Interest
is too obvious to require any illustration.
80. The U3e of the Tables for interchanging Common and
Hyperbolic Logarithms may be learned from the three follow-
ing examples.
(1.) The commfum logarithm of a number being 2.840651, to
find the hyperbolic logarithm of the same number, with nine or
ten decimal figiures.
2 46051701860
.8 1.8420680744
.04 0.0921034037
.0006 0.0013815511
.00005 0.0001151293
.000001 .... 0.0000023026
2.840661 6.5408406471
If the given logarithm has a negative characteristic, we may
proceed as in the following case.
(64) INTRODUCTION.
(2.) To convert the common logarithm 6.89042 into a hyp
boUc logarithm carried to the tenth decimal place.
6.89042= -6.10968
6 11.6129254650
.1 0.2302585093
.009 0.0207232658
.0006 .... 0.0011512925
.00008 0.0001842068
-6.10958 .... -11.7662427394
= 12.2347672606
(3.) The hyperbolic logarithm of a number being 3.0290119,
to find the corresponding common logarithm with ten decimal
figures.
3 1.3028834467
.02 0.0086868896
.009 0.0039086503
.00001 .... 0.0000043429
.000001 0.0000004343
.0000009 .... 0.0000003909
3.0290119 1.3164831537
81. To show the use of the Tabk for finding the lengths of
Circular Arcs the following example wiU be suflScient.
To find the length of an arc of 207^ 41' 38'^09, with twelve
decimal figures.
200° 3.490668503989
70 0.122173047640
40^ 0.011635528347
1' 0.000290888209
30^' 0.000145444104
8'' 0.000038785094
0'^09 0.000000436332
2070 41' 38''.09 . . . 3.624942633716
82. With regard to the two tables on page 340, no explana-
tion is necessary.
'Tf: — Log. *.
I^.
M. Log. A,
W.
-sr
L.S.
3
3oio3oo
477 "13
68
69
71
?3
75
76
78
-if
83
84
85
86
8?
88
89
9'
93
-^
.o3
.04
.o5
iq6
.08
'°9
Hi
114
1.5
1.6
1.8
1^
.33
134
135
i}6
.36
i?9
i3.
.31
i33
8335089
838 8491
85. 3583
857 3335
863 3119
8691317
8750613
88o8i3e
.35
.36
.39
i4i
.43
143
;s
146
.47
.48
.49
i5i
.5i
.53
.54
.55
.56
157-
.58
^
163
ifi3
i65
.66
IS
.69
'7'
173
174
(75
176
178
183
lB3
,84
.85
f
191
Hi
1
.3o3338
.33 5389
3o3
104
305
>o6
S
J09
1.3
1.4
1.5
3.6
3,8
'19
3l3
334
335
316
338
339
^sT
333
333
334
335
336
337
338
^
34.
341
143
144
345
346
347
348
349
157
353
i53
154-
3 55
356
358
359
361
363
363
364
i65
166
3o5 35.4
3074960
309ii3.-,3
3n7539
3.38671
330.463
369
373
974
375
176
379
38.
381
383
384
385
386
387
188
,89
391
III
1
396
3o3
3o3
i'ld
3o5
3o6
^.S
3,^
3i,
3,3
3,3
3,4
3,5
3.6
377
3.8
3,9
l.T
331
3i3
3i5
3i6
1^
3i8
319
HT
333
333
418 ,346
4397533
434^9
436,636
4377506
4393337
4409091
4434798
4440448
4456043
4487063
45o 1491
45.7864
453 3.83
4548449
456 3660
45788.9
4593935
460B978
4638^
465 3819
468"3"4^
4698110
471 39"
473 7564
474 3,63
475 6713
478 5665"
483 8736
4843998
465 73,4
487 1384
488 5507
4899585
49' 7604
494,546
495 5443
499667.
50.0593
5oi4i7'
5037907
5o65o5o
5078559
5091035
5io545o
5.18834
5,31176
5.45478
5i58738
5,7,959
5,98180
5i;.38i
5314443
4
5
6
778i3i3
.36730b
\it:z
155 3360
75836-15-
.6i368o
.64 3539
1673173
17036,7
.73 1863
7
6
9
8450980
^54^
o4i3qj7
<«i8ii
.:s94M
i3
8864907
8910916
8976371
334 3815
3363359
3383796
33o4i38
33, 43B5
3344538
336 4597
3364565
340 444.
3443933
3463S30
348 3n49
3io 34So
35o ,835
354,084
3560359
3579348
359 8355
363 6, 30
365 4880
367 3559
3691,59-
3710679
3739,30
374 7483
3765770
3783979
.4
i5
i6
.461=80
l76o9[3
1041100
908 4850
$758.39
9.90781
■a
178 7S36
914 3793
1816436
.8469,4
.675107
19033.7
193 1346
\$^.
30.3,71
93
31J1193
34M"7
36.7=78
9395.93
9444817
.^93900
968 4819
973 1179
i4
»6
38oaii2
3979400
4.49733
3
'9
4471580
4613980
3^,8?^
3,
3i
33
49136.7
5o5.5oo
5i85i39
53-.1985-
5440680
5563oi5
98677.7
991 3361
9956353
1.4843a
34
35
36
00433.4
0086001
0118371
017 0333
oi5 3o59
1337.65
37
38
3q
5797836
5910646
3339961
i35 5i84
138 o46.
1405493
i43o38o
1455137
4i
43
43
6>3 a4?3
633 4665
o33 4i38
o37 4a65
^333o
0493180
o53 0784
383 0,70
383 8,54
3856.«3
3873V
389,66.
390935,
396,993
3996737
4o, 4oo5
4o3 .3o5
44
1^
643 4537
6531.35
6637578
1479733
35o4)oo
i53 853o
47
48
49
68.=5<3
690.961
o56 9049
0606978
064 4580
C6B.H59
O7i88>o
075 5470
o83 7854
0863598
089905,
09343.7
09601QO
.00 3705
15767B6
16007,4
=6345..
1646178
367 1717
3695139
37.84,6
J74 157H
37646,8
38, c334
383 30T3
385 5573
38780,7
3900346
393 Q56,
3"94466i
3966651
198853.
5.
53
707570.
7160033
714 1759
733-3^-
740 3637
74818B0
755 8749
770 8530
78S3398
7W34o5
8061H00
8.19.34
8195439
54
55
56
59
58
59
67
63
63
64
65
66
404 8337
406 5403
4083400
409933,
4.16.97
4,31998
4i66"4o5"
4,83o.3
4-9 9'557
41. 6^39
4i3 j45g
4148816
M05B97
1,737.3
.30573?
1138516
■ rf.
^ ^- 1
2 LOGARITHMS OP N V M B B R S . 1
N. IjO^. N* IjO«. If. IiOff.
pr.
lAm- if. Jm^. I
334
335
336
337
338
339
34i
342
343
344
345
346
347
348
349
35i
352
353
354
355
356
357
358
359
36i
362
363
364
365
366
367
368
369
371
372
373
374
375
376
377
378
379
38 1
382
383
384
385
386
387
388
389
391
392
393
394
395
396
397
398
399
523 7465
525 o448
526 3393
5276299
528 9167
5301997
4ni
402
4(^3
404
4o5
4o6
407
408
409
4ii
4ia
4i3
4i4
4t5
416
417
4i8
419
4?i
422
423
424
425
426
427
428
429
43i
432
433
434
435
436
437
438
439
44i
442
443
M4
445
446
447
448
449
45 1
452
453
454
455
456
457
458
459
461
462
463
464
465
466
6o3 1444
6d4 2261
605 3o5o
606 38i4
6074550
6085260
467
468
469
471
47a
473
474
475
476
477
478
479
48 1
482
483
485
4«6
'487
488
489
491
492
493
494
495
496
497
498
499
5oi
5o2
5o3
5o4
5o5
5o6
5o7
5o8
509
5ii
5l2
5i3
5i4
5i5
5i6
5i7
5i8
519
521
522
523
524
525
526
527
528
529
53i
532
533
6693169
670 2459
671 1728
534
535
536
537
538
539
541
54a
543
544
545
546
547
548
549
55 1
552
553
554
555
556
557
558
559
56i
562
563
564
565
566
5fi7
568
569
571
572
573
574
575
576
577
578
579
58 1
582
583
584
585
586
587
588
589
?^^
592
593
594
595
596
597
598
599
7275413
728 3538
729 i648
601
602
6o3
604
6o5
606
607
608
609
611
6ia
6i3
6i4
6i5
616
617
618
619
621
622
623
624
625
626
627
628
629
63i
632
633
634
635
636
637
638
639
64 1
642
643
644
645
646
647
648
649
65 1
652
653
654
655
656
657
658
659
661
662
663
664
665
666
778 8745
779 5965
7803173
7810369
781 7554
782 47a6
7831887
7839036
7846173
786041a
786 75i4
787 46cj5
7881684
7888751
7895807
790 285a
7909885
791 6906
793 0916
7937904
7944880
7951846
7958800
79^ 5743
797 2675
7979596
7986506
8000294
8007171
8oi4o37
8020603
802 7737
8034571
8041394
8048207
8o5 5oo9
8o6858o
8075350
808 21 10
8088859
8095597
810 2325
8109043
811 5750
8122447
8i3 58io
8142476
8149132
8155777
8162413
8169038
817 5654
818 2259
8188854
820 201 5
830 8580
821 5i35
822 1681
8228216
823 474a
6730209
6739420
674 8611
7299743
7307823
73i5888
532 7544
534 0261
535 a94r
6095044
6106002
611 7233
675 7783
6766936
677 6070
733 1973
7339993
7347998
536 5584
5378191
5390761
6i3 84i8
614807a
6159301
6785184
679 4279
680 3355
7355989
7363965
737 1926
737 9873
7387806
739 5723
741 i5i6
74 1 9391
7427251
7435008
744 2930
745 0748
745 855V
7466342
74741 18
540 3295
541 579a
5428254
6170003
6180481
6190933
620 i30i
621 1763
62a ai4o
6242821
6253125
62634^4
627 3659
6283889
6294090
63o 4279
63i4438
632 4573
682 i45i
683 0470
6839471
684 8454
685 7417
6866363
687 5290
6884198
6893089
69ro8r5~
691 o65i
6928469
693 7269
694605a
6954817
6963564
6972293
6981005
6998377
700 7037
7oi568o
545 3071
5465427
547 7747
549 o()33
550 2284
55i45oo
552 6()82
553 883o
5550944
557 507a
5587086
5599066
748 9629
749 7363
75o5o84
56i ioi4
562 2929
563 4811
634 4?73
635 4837
636 4879
637 4897
638 4893
639 4865
75! 2791
752 0484
7528164
753 583!
754 3483
755 1 1 23
7566361
7573960
758 1 546
5646661
565 8478
5670264
5693739
570 5429
571 7088
6404814
64 1 474 1
642 4645
702 43o5
703 2914
704 i5o5
5728716
574o3i3
575 1878
644 4386
6454223
6^6 4<>37
705 (X)8o
7o5 6637
7067178
7589119
7596678
7604225
761 1758
76r 9278
7626786
5763414
5774918
578 6392
647 383o
648 3600
649 3349
708 4209
709 2700
710 1 174
5809250
582 o634
5831988
650 3o75
65 1 2780
652 2463
710063!
711 8072
71a 6497
7134905
7143298
7151674
764 1761
764 9230
765 6686
584 3312
585 4607
586 5873
654 1765
655 1 384
6560982
7664128
767 1 55o
7678970
587 71 10
588 8317
5899496
657 0559
658oii4
658 9648
7168377
7176705
7185017
7193313
7201593
720 9857
768 6381
769 3773
770 1 1 53
771 5875
772 3217
773 o547
773 7864
7745170
775 2463
775 9743
7767012
777 4268
592 1768
593 2861
594 3926
6599162
660 8655
6618127
595 4962
5965971
5976952
663 7009
664 6420
665 5810
721 8106
722 6339
7a3 4557
5987905
5998831
6009729
666 5i8o
6674530
6P8 3859
7250945
7259116
726 7272
N.
Los* N. Loi:. N. liOff. N. Lofl;. N. IjOff. |
LOGARI
THMS OF NUMBERS.
3
If. Lost* N* Logr. N. JLo$.
N. Log. N. Log. 1
667
668
669
&71
672
673
674
675
676
677
678
679
681
682
683
684
685
686
687
688
689
69a
693
694
695
696
697
699
701
702
7o3
704
705
706
707
708
J09
711
71a
7i3
714
7i5
716
717
7i8
7'9
721
72a
723
724
725
726
7'7
728
729
73i
732
733
8a4 ia58
8247765
825 4261
734
735
736
737
738
739
74 1
742
743
744
745
746
747
748
749
75i
75a
753
754
755
756
757
758
759
761
76a
76.3
764
765
766
767
768
769
771
772
773
774
775
776
777
778
779
781
782
783
784
785
786
787
788
789
79'
79a
793
794
796
797
798
799
8656961
8662873
8668778
801
80a
8o3
8o4
8o5
806
807
808
809
811
8ia
8i3
8i4
8i5
816
817
818
819
8a I
8aa
823
824
825
826
827
828
829
83 1
83a
833
834
835
836
837
838
839
84 1
842
843
844
845
846
847
848
849
85i
852
853
854
855
856
857
858
859
861
862
863
864
865
866
903 6325
9041744
9047155
905 256o
905 7959
9063350
867
868
869
871
872
873
874
875
876
877
878
879
881
88a
883
884
885
886
887
888
889
891
89a
893
894
895
896
897
898
899
901
90a
903
904
905
906
907
908
909
911
91a
913
914
915
916
9'Z
918
919
921
92 a
.23
924
925
926
927
928
929
93a
933
938019T
938 5197
9390198
940018a
94o5i65
941 oi4a
934
935
936
937
938
939
941
942
943
944
945
946
947
948
949
9bi
952
953
954
955
956
957
958
959
962
963
9^4
965
966
9^7
968
969
971
972
973
974
975
976
978
979
981
98a
983
V84
985
986
987
988
989
991
992
993
994
995
996
997
998
999
970 3469 *
9708116
971 2758
971 7396
972 2028
972 6656
9735896
974 0569
974 5II7
9749720
9754318
975 891 1
9763500
9768083
977266a
978 i8o5
9786.369
979 0929
979 5484
9800034
9S0 4579
9809119
981 3655
9818186
98a 7a34
983 i75i
983 6a63
9840770
984 5a73
9849771
985 4a65
985 8754
986 3a38
987 ai9a
9876663
988 iia8
9885590
9890046
9894498
9898046
9903^89
9907837
9916690
99a 1 1 1 5
99a 5535
9929051
9934362
9938769
9943172
9947560
9951963
8267225
827 3693
828 01 5 1
867 4675
868o564
868 6444
828 6599
829 3o38
8299467
869818a
870 4039
8709888
9068735
9074114
9079485
941 5ii4
94a 0081
94a 5o4i
83o5887
83i 2297
8318698
871 5729
872 1563
872 7388
9090209
9095560
9100905
9106244
911 1576
911 690a
9429996
9434945
9439889
9449759
945 4686
9459607
9464523
946 9433
947 4337
833 1 47 1
833 7844
834 4ao7
835 fo6i
8356906
836 3241
873 3206
873 9016
8744818
875 6390
876 2178
876 7950
91a 2221
9127533
913 2839
836 9567
837 5884
838219a
8773713
877 9470
8785218
914343a
914871a
9153998
947 9236
948 4i3o
9489018
9498777
950 3649
95o85i5
951 3375
95i 823o
9523080
839 4780
840 1 061
840733a
879 0959
8796692
8802418
881 3847
881 9550
88a 5a45
915927a
9164539
9169800
84i 3595
84 1 9848
84a 609a
917 5o55
918 o3o3
918 5545
843 a3a8
843 8554
844477a
8830934
883 6614
884 aa88
9196010
920 1233
920 645o
952 7924
953 2763
953 7597
8457180
846 3371
846 9553
8847954
885 361 a
8859263
9ai 1661
9a I 6865
9a a ao63
954 7248
955 2o65
955 6878
847 57a7
848 1891
8488047
8870544
8876173
888 1795
9aa 7a55
923 a44o
923 7620
9561684
9566486
957 128a
957 6073
958o858
958 5639
9595184
9599948
960 4708
8494194
85o o333
85o646a
8887410
8893017
8898617
9247960
9253121
925 8276
8518696
85a48oD
8530895
8904210
926 3424
9268567
927 3704
853698a
854 3o6o
854 9i3o
8926510
8932068
8937618
927 8834
928 3959
9289077
9609462
961 4211
961 8955
855519a
856ia44
856 7289
8943161
8948697
8954225
929 9^96
930 4396
930 9490
962 3693
962 8427
963 3i55
857 9353
858 537a
8591383
8597386
86o338o
8ex>9366
895 9747
896 5a62
897 0770
931 4579
931 9661
932 4738
964 2596
9647309
965 2017
6981765
8987252
899 273a
899 8ao5
9003671
9009131
9329808
933 4873
933 993a
965 6720
9661417
9666110
9670707
9675480
9680157
9960737
9965117
996949a
997 3864
9978231
998 3593
998605a
999 iJo5
999 5655
861 5344
86a i3i4
862 7275
935 oo32
935 5073
9360108
8639174
864 5111
865 io4o
901 4583
9oaooa9
90a 5468
936 5 1 37
937 0161
9375179
9689497
9694159
9698816
W.
Jjog* N* Vog* N« Log. N. Log. N. Log. |
LOOABITHMS OF NUMBERS.
Nnnu 12 3 4
1000
01
oa
o3
o4
o5
o6
07
08
09
1010
II
la
i3
i4
i5
16
t?
18
»9
10 ao
31
aa
a3
M
a5
a6
27
28
39
10 3o
3i
3a
33
34
35
36
3?
38
39
to4o
4i
4a
43
44
45
46
47
48
49
10 5o
5i
5a
53
54
55
56
57
58
59
Num.
OOO0000I0434
4341 4775
91 1 1
344a
7770
• 8677
001 3009
• 7337
00a 1661
5980
oo3oa95
46o5
• 891a
oo43ai4
• 7512
oo5 i8o5
6004
oo6o3oo
4660
• 8937
0073210
• 747S
008 174a
6002
0090357
4509
• 8756
oio3ooo
• 7339
on 1 474
5704
• 9931
oia4i54
• 8372
0133587
• 6797
oi4 ioo3
SaoS
• 94o3
0153598
• 7788
016 1974
61 55
0T70333
4507
• 8677
018 a843
• 7005
0191163
53i7
• 0467
oao j6i3
•' 7755
oai 1893
6037
oaaoi57
4a84
• 84o6
0869
5ao8
q544
3875
8aoa
r3o3
5642
9077 #41 1
1737
6076
2093 252^
64II6843
0726 1 1 57
5o36 5467
934a
3644
7941
2234
65a3
0808
5o88
9365
3637
7904
2168
6427
o683
4934
9181
34a4
766a
1897
6137
♦354
4576
9773
4074
837 T
3663
6953
1336
55i6
9702
4064
833i
2594
6853
1108
5359
9603
3848
8086
a33o
655o
0776
4998
87949315
3oo8
7318
i4a4
5635
9823
4017
8206
2392
6573
0751
4924
74a I
1578
5732
988a
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Die iL Mah.
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5
6
7
8
1
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3
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6
7
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6
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LOGARITHMS OF NUMBERS.
Nam.
106O
6i
63
63
64
65
66
67
68
69
1070
7«
72
73
74
75
76
77
78
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81
83
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84
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86
87
88
89
1090
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92
93
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026 i?45 1 654
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6
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3102
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10
LOGARITHMS OF NUXBBES.
um. 12 3 4
N
1950
• 6i
62
63
64
65
66
67
68
69
1370
71
72
73
74
75
76
77
78
79
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82
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86
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133 538915708
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4959 5277
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5976
8766
1 534
43ii
7086
9868
263o
8167
2210
4948
7684
0419
162
5883
8612
1 340
4066
6790
9612
2233
4?52
7070
a385
7463
♦280
3096
10
8723
1 533
4342
7148
2757
5558
8357
ii55
3961
6745
3473
6254
9034
1812
4588
7363
♦ i36
2907
5676
8443
3973
6735
5oiT
7766
0620
3271
60a I
8769
i5i6
4260
7003
9745
2484
5222
7958
0692
3425
6i56
8885
1612
4338
7062
9785
25o5
5224
0667
7745
0662
3378
6192
9004
1814
4622
7429
♦234
3o37
5838
8637
1435
423f
7025
9817
2607
5396
8i83
0968
3761
6532
9312
2090
4866
7640
o4i3
3i84
5953
8720
i485
4249
7011
9771
253o
5287
8042
0795
3546
6296
9044
1790
4535
7278
♦019
2768
5496
823i
6420
91 58
i885
4611
7335
♦067
2777
5496
8213
0928
01234 56789
DiflT. k, MvU.
1
281
2
562
3
843
4
1124
5
i4o5
6
1686
7
1967
8
2248
9
2629
I
2
3
4
5
6
7
8
9
I
2
3
4
5
6
7
8
9
I
2
3
4
5
6
7
8
9
279
558
837
1116
1395
1674
1953
2232
25ll
273
546
819
1092
i365
i638
1911
2184
2457
282
564
846
1 128
i4io
1692
1974
2256
2538
280
56o
84o
1 1 20
i4oo
1680
i960
2240
2620
I
277
2
554
3
83i
4
1108
5
1 385
6
1662
7
8
1930
2216
9
2493
278
556
834
1112
1390
1668
1946
2224
2602
275
5-5o
825
IIOO
1375
i65o
1925
2200
2475
276
552
828
iio4
i38o
i656
1932
2208
2484
274
548
822
1096
1370
1644
1918
2192
2406
I
271
a
542
3
8i3
4
1084
5
i355
6
1626
7
8
1807
2168
9
2439
272
544
816
1088
i36o
J 632
1904
2176
2448
Diir. & Malt.
B
14
LOGABITHMS OF NUMBERS.
Nam.
1000
oi
oa
o3
o4
o5
06
07
08
09
16 10
II
17
i3
i4
i5
16
17
18
19
1620
at
aa
23
24
a*)
a6
27
28
29
16 3o
3i
32
33
34
35
36
37
38
39
1640
4i
42
43
44
45
46
47
48
49
16 5o
5i
5a
53
54
55
56
57
58
59
Nam.
12 3 4
5 6 7 8
9
3542"
6354
9064
1773
4480
7i85
9889
2590
520I
79B9
0686
338i
6074
8766
1 456
Diir. Jc Malt.
60:
204 1200
i3
25
• 9335
2o5 2044
475o
7455
2060159
2860
556o
• 8a59
ao7 095^
365<)
6344
• 9035
208 1725
44x4
7100
ao9 3468
5i5o
• 783<)
aioo5o8
3i85
586o
• 8534
an i2o5
38-6
6544
• 92 1 1
212 1876
4540
7203
# 9863
2i3 a52i
4839
747'
2180100
372
5355
I
• 7980
2190603
3aa5
5845
1471
4i85
6896
96c>()
33i4
5oai
7736
0439
3i3i
583o
8520
123$
3930
6(>i3
9304
1904
4()82
7369
♦o54
2737
54i8
8098
0776
3453
6128
8801
1472
4i42
6811
9477
3l42
4806
7468
♦ 128
2786
5443
8098
0702
3404
6o55
8703
i35o
3096
6640
9282
1923
4562
7200
9836
2470
5io3
7734
o363
2991
56i8
8243
0866
3487
6107
1743
4456
7167
9877
2585
5292
7996
0699
3401
6ioo
8798
1495
4189
6882
9573
2263
4951
7<>37
o333
20l4
4727
7438
♦i48
2856
5562
8267
0969
3671
6370
9068
17^)4
4459
7i5i
9842
2532
5220
7906
0390
3oo5 3273
5686
836()
1044
8464 8726I8987
5954
8634
l3l3
3730 3988
6395 6662
9068
1740
4409
7078
9744
2409
5072
7734
039410660
3o52 33i8
9335
2007
4676
7344
♦on
2675
5338
8000
5709
8364
1017
3669
63 1 9
6968
i6i5
4260
6904
9546
2187
4836
7463
5366
7997
0636
3354
588o
85o5
n28
3749
6369
?)974
8629
1283
3934
6584
9233
1880
4525
7169
98n
245 1
5090
7727
o363
2997
5629
8260
0889
35i7
6143
8767
1390
4oM
663 1
9249
2285
4998
7709
0419
3127
5833
8537
1240
I
o
9338
2o34
4728
7421
♦ ni
2801
5488
8174
c^59
3541
6322
8902
579
4255
t)93o
9603
3274
4043
7on
0277
2942
5()o5
8266
0926
3584
6240
8895
i548
4199
6849
9498
3i44
4780
7433
♦075
2715
5354
0026
3260
5892
8523
n52
3779
64o5
9o3o
1 652
4273
6893
95n
2557
5269
7980
06900960
33973668
6io3
8807
i5io
42n
6910
9607
23o3
4997
7690
o38o
3070
5757
8443
1 1 27
38io
6490
9170
1847
45a3
7197
9870
a54i
5210
7878
o544
3208
5871
8532
n9r
3849
65o5
9160
i8i3
4464
7»i4
9762
2409
5o54
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0339
2979
5617
8254
0890
3523
6i55
8786
i4i5
4043
6668
9292
1914
4535
7i55
9773
2828
5541
825 1
6374
9078
1780
4481
7180
9877
3573
5267
7959
06490918
3338
6026
871 1
1395
4078
6758
9437
2n5
479«
7464
♦i37
2808
5477
8144
0810
3474
6i37
8798
1457
4n5
6771
9425
2078
473o
7379
♦027
2673
53i8
7961
o(x>3
3243
588 T
85i8
n53
3786
64i8
904993
1677
43o5
6930
9554
2177
4797
7417
♦o34
58ia
8S22
[23l
3939 4209
6644
9348
2o5o
475i
7449
♦ I 47
2843
55.36
8228
36o7
6294
8980
i664
4346
7026
9705
2382
5o58
7732
o4o4
.3075
5744
841 1
1077
3741
64o3
9064
1723
438 1
7037
9691
2343
4995
7644
0292
29S8
5583
8226
0867
3507
6145
8781
i4i6
6682
12
1940
4567
7193
9816
2439
5o59
7678
0296
3371
6o83
8793
l502
6915
9618
2320
5o2i
7719
o4i6
3lI2
58o5
8407
n87
3876J4
6563
9248
1932
46i4
7294
9973
265o
5325
7999
0671
3342
6010
8678
1 343
4007
6669
9330
4646
7302
9956
3(x>9
5260
7909
o556
32o3
5847
84QO
n3i
3771
6409
9045
1680
4o5o43i3
6945
9575
2303
483(>
7455
♦079
3701
53a!
7940
o558
145
6832
9517
2200
488a
7562
♦ 24l
2918
5593
8266
3609
6277
8q44
lOIO
4273
6935
4912
7568
♦ 22T
2874
5535
8174
0821
3467
6ni
8754
1395
4o34
(1672
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4576
7208
9838
2466
509a
7718
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5583
8202
0819
I
a
3
4
5
6
7
8
9
I
a
3
4
5
6
7
8
9
01234 56789
I
a
3
4
5
6
7
8
271
542
8r3
1084
i355
1626
1807
ai68
2439
373
544
816
1088
i36o
i633
1904
2176
2448
I
260
a
538
3
807
4
1076
5
i345
6
1614
7
i883
8
ai5a
9
2421
263
526
789
io52
i3i5
1578
i84i
aio4
2367
261
522
783
1044
i3o5
i566
1827
2088
2349
270
540
810
ro8o
i35o
1620
1890
aioo
343o
I
267
2
534
3
801
4
1068
5
1 335
6
1602
7
8
1869
2i36
9
24o3
268
536
8o4
107a
1 340
1608
1876
2144
3412
I
265
2
53o
3
4
795
1060
5
i325
6
1590
7
1 855
8
2120
9
2385
366
532
798
IC164
i33o
1596
1862
3128
2394
364
538
793
io56
1 320
i584
1848
2na
3376
363
524
786
io48
i3io
1573
i834
3096
3358
Diff. It Mult.
LOGARITHMS OF MnMBBRS.
IS
H«.
12 3 4
5 6 7 8 9
OUT.*
Hull.
lM0[23O
jo8.|iJ43
1604
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3913,3.7313435
360
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4481
4743
5365
553615788
6049
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530
5"
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7617
7878
8,396400
866,
3
780
783
9184
944s
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0489
4
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1044
■794
3o55
33i6
3838
3<W
3360
363,
3883
5
i3oo
i3o5
44o3
7011
Si
%l
5i86
7793
5446
8o53
5707
83t4
sfS
%1
548|
6
7
i56o
1830
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4834
5o84
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3741
5344
3oo3
56o4
o658
3=63
5864
5134
378?
6384
,440
4043
6645
43I3
6905
8
9
ao8o
3340
3088
3349
7435
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7665
SS
B3o5
0804
8465
1064
B735
.334
%
9145
,843
95o5
3103
1631
1883
3.43
3403
366i
3j31
4.8,
4440
4700
Si?
B073
5738
8333
IS
6q57
885a
65i7
9111
6776
7o36
9630
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358
5,6
a
0407
0667
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1444
1704
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348.
774
777
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3358
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3777
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5073
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6366
6635
7143
766.
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18,3
8178
0766
8437
1034
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%
973,
33.7
3834
6
I
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1548
1806
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4127
4385
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4903
5,6o
5419
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6194
6453
6710
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7485
7743
8003
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9034
9,93
9B09
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1099
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1873
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3647
3905
3,63
3679
4.94
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73B8
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9633
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L06AB1THMS OF NUMBERS.
23
Num.
3140
4i
43
43
44
45
46
47
48
49
31 5o
5i
5a
53
54
55
56
58
59
31 6o
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63
63
64
65
66
67
68
69
3170
71
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73
74
75
76
77
78
79
31 80
81
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83
84
85
86
87
88
89
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01234 56789
330 4i38
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m
24
LOGARITHMS OF NUMBERS.
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3200
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06
07
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L06ABITHMS OF NUMBERS
25
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356 oa
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6048*6239
81 54
♦067
1789
3701
56i2
75a3
9432
i34i
3249 3440
5i56
2
9341
1232
3l22
5oi2
6901
0670
2562
4448
6332
8216
743676247812
95o3
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3258
5i35
7010
8885
0760
2633
45o5
6771
8672
0572
2472
4370
6268
8i65
♦061
1957
385i
5745
7638
9530
1421
33ji
520I
7090
8977
o865
2751
4636
6521
84o5
0288
2170
4o5i
5932
7198
9073
oo47
2820
4693
IXff.
192
191
190
189
188
Difll
Difll St Moh.
1
2
192
384
3
576
4
768
5
960
6
1 1 52
7
1 344
8
1 536
9
1728
I
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2
382
3
573
4
764
5
955
6
ii46
7
1 337
8
1 528
9
1719
I
2
38o
3
570
4
760
5
950
6
ii4o
7
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8
l520
9
1710
I
180
2
378
3
567
4
756
5
945
6
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1701
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188
2
376
3
564
4
752
5
940
6
1128
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8
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9
1692
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187
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374
3
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4
748
5
935
6
1122
7
i3o9
8
i4q6
9
1 683
I
DifRAtMolU
26
LOGARITHMS OF NUMBBBS.
Nam.
2330
31
aa
a3
a5
26
27
a8
a9
a3 3o
3i
3a
33
34
35
36
37
38
39
2340
41
42
43
44
45
46
47
48
49
23 5o
5[
52
53
54
55
56
57
58
59
23 60
6r
6a
63
64
65
66
67
68
69
2370
71
72
73
74
75
76
77
78
79
Nam.
365 4880
6751
• 862a
366049a
a36i
4a3o
2869
4728
6587
• 8445
369030a
ai59
4oi4
586o
77a3
• 9576
370 i4a8
• 3a8o
5i3i
6981
• 883o
3710670
2520
4373
6ai9
8o65
37a 175
m
7379 7464
• 91 ao
3730960
6475
83ii
3740147
1983
38i7
565i
7483
• 9316
375 1 1 47
2077
4607
6636
• 8464
376 029a
aii9
3944
5067
0679
a54d
44i6
6a84
8i5u
♦016
1881
3746
5609
7472
9334
1 195
3o55
4914
6773
8631
o488
7344
4200
6o54
7908
9761
i6i4
3465
53i6
7166
90i5
o863
3711
4558
64o4
ii44
3983
4821
6658
84q5
o3Ji
ai66
4000
5834
7667
n^
3i6o
4990
681
8647
0475
a3oi
4127
2
52?4
7ia6
597
2735
46o3
6471
8337
oao3
ao68
393 a
5795
7658
95ao
i38i
334t
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6959
8817
0674
253o
4385
6240
8094
9947
365o
55oi
735i
9300
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3896
4742
6588
8a49 8434
♦004
1937
3780
56a a
0378
3133
3q64
58o6
7^48
93049488
1 338
3167
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4i84
6017
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968a
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5173
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9183
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3933
5 6 7 8 9
4790
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0389
3354
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5982
7844
9706
i567
3437
5386
145
9003
0859
3915
4571
6425
8279
♦i3a
1984
3835
5686
7536
9385
1333
3o8o
4927
6773
8618
0463
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4149
7833
9673
l5l2
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5189
7036
8863
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2533
4367
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8o33
9865
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35a6
5356
7185
9013
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366638
449a
56a9
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9370
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3109
4977
6844
8710
0576
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6168
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1753
36i3
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9188
1045
3901
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7721
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0647
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4333
6175
8016
9856
1696
3534
5372
7310
9046
0883
3716
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6384
8a 16
♦048
1879
3709
5539
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9195
1033
467
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58i6(6oo3
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1939
37
56
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9754
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8987
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2674
4517
6359
8300
♦o4o
1870
3718
5556
7393
9230
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3900
4734
6567
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5733
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4857
7874
9744
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535o
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4677
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7703
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4171
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7888
9745
1603
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3735
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6436
8375
♦ 134
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5666
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1 300
3o43
4885
6737
8568
O334|o4o8
3o63
3903
5740
7577
3347
4086
5924
7761
9597
1 43a
3267
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6934
7916
9744
1 571
3397
5333
6377
8348
♦ 118
7
56
198
385
5734
7591
9457
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4357
631 5
8074
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9305
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6564
8435
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8359
♦117
1973
3839
5683
7538
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1243
3095
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6796
8645
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2342
189
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354
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3i57
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7696
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3337
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691 1
8753
05030776
34^1 361 5
4370 4453
6107 6391
79448138
97809964
1616
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5284
7117
876689499133
0598 0781 0964
3438 361 1 3794
4358 444i 4634
6088 6370 6453
8090
0036
993
1753
357
540
ll
3634
5467
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8383
♦ 109
1936
3763
5587
DHL
I«7
DifffcHolL
186
i85
184
i83
I
3
3
4
5
6
7
8
I
a
3
4
5
6
7
8
9
X
3
3
4
5
6
7
8
I
3
3
4
5
6
7
8
9
t
3
3
4
5
6
7
8
X
a
3
4
5
6
7
8
9
188
376
564
75a
940
1138
i3i6
i5a4
1693
t86
373
558
744
930
iir6
l303
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1674
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370
555
740
925
mo
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184
368
553
736
930
iio4
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1472
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366
549
73a
915
ioq8
1 301
1464
1647
01234 56789
Dir.
HitL^MvOu
LOGARITHMS OP MITMBBRS.
27
85
86
87
88
«9
3390
9»
9a
93
p
99
24 00
01
02
o3
04
o5
06
07
68
09
II
la
i3
U
i5
16
17
18
'9
3490
21
23
23
24
25
26
27
28
29
24 3o
3i
32
33
34
35
36
37
38
39
3T6 5770 5952
fimm, 1
2380
81
82
83
84
7594
• 9418
377 1240
3o63
4884 5o66
5248
6704)68867068
# 85248706
378 o343 o525
2 161 2343
f
7612
• 9^27
379 I24t
3o55
79416
96597
4868 5o49 523
66806662
• 8492
38oo3o2
2293
4l02
II
7718
9345 952S
2112
3922
5730159
7538
38i ii5i
2956
4761
6565
8368
3820x70
1972
3773
5573
7373
• 9171
3830969
2767
4563
81 54
• 9948
384 1741
3534
5326
7117
6o63
785o
• 9636
386t42i
3206
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6773
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3870337
2118
7776
96009782
Q Q
6i35«3t7
7959 8i4i
5 6 7 8 9
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3245
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9608
1423
3237
8673
o484
8888
0707
2525
4342
6159
7975
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3418
7043
8854
o665
i33i
3i37
494t
6745
8548
o35i
21 52
3953
5-753
7553
2474
4283
6092
7899
9706
l5l2
33i7
5l22
6926
9351
ii4q
2946
4743
6359|6538
8333
♦127
1921
3713
55o5
7207
9087
0877
2666
4454
6241
8oa8
9814
1600
3384
5i68
6951
8733
o5i5
2296
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3427
873989099089
o53i
233a
4i33
5933
7732
9531
3126
4922
6718
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o3o7
2100I
7476
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2845
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6420
8207
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1778
3563
5346
7129
^\
0693
2474
9965
1787
360913791
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9070
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2707
4524
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81 56 8338
1786
35993780
5412
7224
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9887
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♦067
1693
3498
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92669445
1386
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3o23
4812
83
♦171
1957
3741
5525
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9090
0871
2652
♦147
1969
56i2
7432
9a 52
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2889
4706
6522
♦I 53
1967
5593
74o5
216
1027
2836
1873
3678
5483
7286
0891
3693
4493
6393
809a
0711
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43i3
6ii3
7912
97"
t5o9
33o6
5ioa
6897
869a
0486
2279 ^4^9
4072
58646043
3485
5281
7077
8871
o665
7834
9624
i4i3
3 203
4990
6778
8564
o35o
3i35
3919
5703
7486
9268
1049
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6682t6864l7o47]73
85o6
0329
2l52
39734155
5794
7614
9434
1353
3070
4887
6704
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2i48
987
3962
5774
7586
9397
1208
3oi7
4826
6634
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♦070
1868
3665
5461
905 1
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6322
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9803
1592
3381
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3a52
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7647
9369 9450 9630 98
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7a56 7436
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996
1771
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5348
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3a33
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43oo6
o
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3844 4034 4304
5820
7615
10
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46094789
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♦161
1950
3739 391
944696349803
13381406 1 584
3oo8 3i86 3364
5527
73i4
7i35
8931
0707
7492
42764455
60606238
7412
9235
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3880
4702
6522
8342
♦161
1980
3797
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743o
9245
1060
2874
45064687
905
0876
2698
4520
634o
8160
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1798
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5432
7349
9064
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3693
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1750
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5368
7176
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0790
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10
1612
341 3
5ai3
701 3
8813
10
2407
6000
7795
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6499
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♦121
1931
3741
5549
7357
9164
0970
2776
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6384
8f88
9990
1792
3593
5393
7193
8992
0790
2587
4383
6179
7974
9769
1 56a
3355
5i47
3176
4968
6759I6938
8739
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4096
5884
DMC
9100
0886
2670
5705
749317671
9279 9457
803 1
ro64
2849
1243
3027
4633 481 1
64166595
8109
9961
1762
3543
8377
♦ 159
1940
3720
01234 56789
183
181
180
179
178
Die It Hah.
I
3
3
4
5
6
7
8
9
I
2
3
4
5
6
7
8
9
I
2
3
4
5
6
7
8
9
I
2
3
4
5
6
7
8
9
Dur.
I
2
3
4
5
6
7
8
9
I
a
3
4
5
6
7
8
9
i83
366
549
73a
915
1098
laSi
1464
1647
i8a
364
546
728
910
1092
1274
1456
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x8i
362
543
734
905
1086
1267
i448
1629
180
36o
540
730
900
1080
1260
i44o
i6ao
170
358
537
716
895
1074
1253
1433
1611
178
356
534
713
890
1068
1246
1424
i6oa
Diir. «& Molt.
28
LOGAEITBMS 07 IfUKBBRS.
3440
4i
42
43
44
45
46
47
48
49
a45o
5i
5a
53
54
55
56
57
58
59
2460
61
6a
63
64
65
66
67
68
69
2470
71
7a
73
74
75
76
77
78
79
a48o
81
8a
83
84
85
86
87
88
89
a49o
9>
9a
94
96
98
99
Num.
387 3898
5678
7457
• 9a35
388 loia
3789
4565
634o
8ii4
• 9888
38oi66i
3433
5ao5
6975
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390051 5
2284
4o5a
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758
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391 1116
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6407
8169
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392 1691
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521 1
6970
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3930485
2241
3997
5752
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3941013
2765
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6268
6018
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395 i5i6
3264
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6758
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It
5480
7223
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2446
4i85
12
4076 4254
5856
7634
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1190
2966
474a
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1 838
36io
5382
7153
0693
2400
4228
9528
i2o3
3o57
4830
6583
8345
♦ 107
1868
3628
5387
7145
0660
2417
4172
5928
7683
9435
1188
2940
4692
6443
8193
1691
343o
5i86
6932
8678
0423
2168
3912
5655
73o7
9139
0880
2670
435o
201 5
3787
5559
733o
9100
0869
2637
44o5
6172
7939
9704
1469
3233
6760
85a2
0283
2044
38o3
5563
7321
2592
4348
6io3
7857
9611
1 364
3ii
3
445^
6ai2
1 545
33a I
5097
6872
8646
0420
ai93
3965
5736
7507
9276
1046
2814
4582
6349
8ii5
9881
1646
3410
5173
6936
4867
6618
8368
♦ 117
1866
36i3
536i
7107
8853
0598
a 342
4086
4
4610
6389
8168
9946
1723
7o5o
8834
0597
3370
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5913
7684
9453
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2901
4759
65a5
8293
♦057
1833
3586
5349
711a
8874
9|o635
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4i55
5914
63
8608
0459
2230
7497
9355
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6278
8o33
9786996
1539
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5o42
6793
8543
0292
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3788
5535
7282
9027
0772
25l7
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58296003
592416090
766317836
7571
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6272
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1 187
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8 1 8418358
4788
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♦ 133
1900
3677
5453
7337
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6745
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96309807
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6
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324
486
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810
972
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1296
1 458
Diff. Ac MqK.
32
LOGARITHMS OF NUMBERS.
Num.
3680
8i
83
83
84
85
86
87
88
89
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93
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16
17
18
19
2720
21
22
23
24
25
26
27
28
29
2730
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33
34
35
36
37
38
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01234 56789
438 1 348
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12 3 4
5 6 7 8 9
Diir.
125
124
DiflT.
Diff.lbMiilt.
1
3
3
4
5
6
7
8
I
a
3
4
5
6
7
8
9
X
3
3
4
5
6
7
8
9
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3
3
4
5
6
7
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353
378
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63o
756
883
1008
1x34
135
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375
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625
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875
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124
348
373
496
630
744
868
993
1 1 16
133
246
369
493
61 5
738
861
984
X107
Difl*. & Malt.
46
LOGARITHMS OF NUMBEBS.
Num.
3530
ai
a3
a4
25
26
27
28
29
35 3o
3i
32
33
34
35
36
37
38
39
3540
4i
42
43
44
45
46
47
48
49
35 5o
5i
52
53
54
55
56
57
58
59
35 60
61
62
63
64
65
66
67
68
69
3570
71
72
73
74
75
76
77
78
79
Nanu
12 3 4
5 6 7 8 9
646 5427 555o
6b(yo 6784
78948017
• 91269250
547 0359 0482
1 591
2823
4o55
5i86
65i7
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2665
3894
5i23
635 1
7578
8806
549 oo33
1269
2486
3712
4937
6162
7387
8612
• 9836
550 1060
2284
35o7
473o
5962
7174
8396
• 9618
55 1 0839
2059
3280
45oo
5720
6939
81 58
• 9377
552 0595
i8i3
3o3i
4248
5465
6682
7899
• 9115
553 o33o
1 545
2760
3975
5189
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2946
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4017
5245
6473
7701
8928
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5673
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5533
6763
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9233
0453
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8734
9959
1182
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3629
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6074
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0961
2181
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5842
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2608 3731
3957
5182
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♦081
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2528
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9862
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1 327
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7 1 32
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1068
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2118
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6270
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8731
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Difl:
3384
4607
583o
7o52
8274
9495
0717
1937
3i58
4378
5598
6817
8o36
9255
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3854
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7496
8709
123
laa
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2
3
4
5
6
7
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3
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5
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9
348
373
496
630
744
868
992
1x10
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738
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984
1 107
01234 56789
Dir.
t
laa
a
344
3
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4
488
5
6to
6
73a
7
854
8
9^
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Dur.*;
LOGARITHMS OF ICnTHBEES.
47
1 2
8i
8a
63
84
85
86
87
88
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3590
9"
%
99
36 00
01
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08
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la
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18
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ai
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33
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35
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38
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5S3 883oj895a
554 0043 0164
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Dir.
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9778
0921
2o63
32u4
4346
')487
6627
7768
8908
♦o48
1 187
2326
3465
44904604
574a
6880
8018
9155
0293
a6oa
3957
5iii
6a65
a687
384a
6i5o
73o4
8457
9610
0763
1916
3o68
4aao
5371
652316638
2918
407a
5227
638i
7419 7534
8573 8688
97269841
0878
2o3i
3i83
4335
5487
635o6464
7607
8750
9803
io35
a 1 77
33i8
446o
56oi
6741
7882
9oaa
oi6a
i3oi
3440
357
471
5856
6904
8i33
o4o6
0994
3i46
3398
445o
56oa
6753
7904
7789
89399655
♦090
ia4o
3390
3539
4688
5837
6986
8i34
9383
0439
1577
3734
3870
5oi7
6i63
7309
8454
9599
0744
1889
3o33
4177
5331
oao5
4355
a5o5
3654
48o3
595a
7100
8349
9397
o544
1691
3838
3985
5i3i
6278
7423
8569
9714
o85o
200d
3i48
4292
5435
6579
7722
8864
♦007
"49
2291
34J3
4574
5715
6855
9^
0276
i4i5
3554
483a
8483
5970
7108
8a45
9383
o5ao
01234 56789
ii5
xi4
I
a
3
4
5
6
7
8
9
I
a
3
4
5
6
7
8
Diir.
I
a
3
4
5
6
7
8
116
aSa
34B
464
58o
^
8ia
9a8
1044
ii5
a3o
345
460
575
690
8o5
920
io35
114
aa8
34a
456
570
684
798
91a
ioa6
DiC^IML
LOGARITHMS OF KUMBBRS.
51
12 8 4
5 6 7 8 9
ai
aa
a3
aS
a6
a8
39
38 3o
3i
3a
33
34
35
36
37
38
39
38 4o
4i
42
43
45
46
47
48
49
38 So
5[
5a
53
54
55
56
57
58
59
38 6o
6i
6a
63
64
65
66
67
68
69
3870
71
72
73
74
583 0634
1770
2907
4o43
0747
1884
3oao|3
4i57
5179 5a9a
63i4
745o
8585
• 9719
583o854
1988
3iaa
4a55
5388
65a I
7654
8786
• 9918
584io5o
ai8i
64a8
7563
0967
atoi
3a35
4368
55oi
16634
33ia
4443
5574
34a5
4556
5687
&7046817
7634 7947
8963 9076 9189 93<
5850093
laaa
a35i
46cy7
5735
6863
• 91 17
586oa44
1370
2496
36aa
4748
5873
8ia3
• 9247
5870371
1495
a6i8
374a
4865
5987
7110
8a3a
• 9353
588 0475 o587
1708
0861
3i34
4270
5406
654i
7677
88ia
9946
1081
aai5
3348
448a
56i5
6748
7767
8899
♦oBi
ii63
aa94
oao6
1 335
a463
3479 3592
4720
5848
6976
79908103
1607
373 1
3854
4977
6100
8344
78807993
9ia6
oa58
1389
a5ao
901a
0144
1376
2407
3^38
4669478
58oo59
6930
8060
o3i9
1448
3576
3705
4833
5961
7088
8ai6
93309342
o356
i483
3609
3735
0469
1596
3733
3847
48604973
59866098
7110
8335
9360
o484
7333
8348
9472
0596
1730
3843
6313
733a 7334
0975
31 1 1
3348
43844497
5530 5633
6655^769
7790
8935
♦060
1194
33a8
346a
4595
5728
6861
3653
3
i3
7^3
8173
1088
3335
336i
03
o43a
i56i
3689
38i8
7904
9o3o
0173
i3o7
a44i
3575
4708
584i
6974
8107
9339
0371
l503
3634
3765
4895
6036
7i56
8a86
94i5
o544
1673
380a
3930
4946 5o58
6673
7301
83a8
9455
o58a
1708
3834
5o85
6aii
7335
9584
0708
i83a
3955
1596
75
76
77
79 I 71981731^0
2717
3838
4958
6078
94669578
0699
i8ao
3839
56705
6190
39664079
50895301
6334
7446
8566
8456
2941
39564663
i8a
63oa
74a a
6186
73i4
844i
9568
0694
i8ai
3947
3960407a
5198
6333
7448
84608573
9697
0831
1944
3o68
4191
53i4
6436
7559
8680
96909803
081 1 0933
1933 ao45
3i65
4386
54o6
65^6
7646
3o53
4174
5994
64t4
7534
I303
a339
3475
4611
5747
688a
8017
91 5a
oa87
l43I
a555
3688
48aa
5955
7087
8aao
9353
o484
i6i5
2747
3878
5oo8
6139
7269
3991
5l3I
6353
7383
8399851a
953
o657
1786
3915
4043
5171
6399
74ao
8554
9681
0807
1933
3o59
4i85
53io
89641
7560
8685
7671
8793
io36
ai57
3377
4398
55i8
6638
I T758
3453
3589
4725
586o
^
9365
o4oo
1534
3668
38o3
4935
6068
7301
8333
9465
0597
1739
3860
14291
3566
3703
4838
07700883
3o38
4i56
5384
64i3
7539
8666
9793
0930
3o46
3173
4398
5433
64366548
7673
8797
993a
io45
ao57 3169
31803393
43o3
5436
65496661
7783
8905
♦036
ii48
33
45io
563o
6750
i543
a68o
38i6
4953
59746087
71097333
8344"
o5iS
i648
3781
3915
5o48
6181
73i4
8446
9578
0710
i84a
2973
4io4
5334
6365
7495
8635
975498679980
0996
3135
3a53
426914383
30 1 a
3i4i
5397
6535
7653
8770
9906
io33
3i59
3385
44io
5535
6661
7785
8910
♦o34
ii58
3381
34o5
44i6|4538
5538 5651
6773
7895
9017
01
I36o
693381
3389 35o
a
4633
5742
6863
787017981
8358
9492
0637
1761
3895
4038
5i6a
6395
7427
856o
^.
0833
1955
3o86
4217
5348
6478 659
7608
8738
6637
7765
889a
♦019
ii45
3371
3397
45a3
5648
6773
7898
90a a
01 46
1370
3394
35i7
5763
6885
39035
8007
91a
I
i373
3493
36i4
4734
i657
2793
3929
5o65
6301
7336
8471
9606
0740
1874
3oo8
4x42
5375
6408
7540
8673
9805
0937
3068
3199
433o
5461
I
7721
885o
1109
3338
3366
4494
55io56a3
6750
7877
9004
oi3i
1358
7384
35io
4635
5761
6886
8010
9i35
oa59
1 383
a5o6
3639
464o475a
5875
6997
81 ao
I
6363
1484
a6o5
9934
3726
4846
585415966
69747086
80938305
01234 56789
Diff.
ii3
IJ9
Diir.
Diff.Jkll«lt.
I
3
3
4
5
6
7
8
I
a
3
4
5
6
7
8
9
I
3
3
4
5
6
7
8
9
I
3
3
4
5
6
7
8
114
338
342
456
570
684
798
913
1036
ii3
aa6
339
45a
565
678
791
904
1017
iia
aa4
336
448
56o
673
784
1008
III
333
333
444
555
000
777
888
999
DUr. k. Muh.
52
LOGARITHMS OP NUXBBRS.
Nwn.
8l
Si
83
84
85
86
»7
88
89
3890
9^
93
p
99
3900
01
oa
o3
04
o5
06
07
08
09
39 10
II
12
i3
i4
i5
16
17
18
»9
3930
21
22
23
24
25
26
27
28
29
3930
3i
32
33
34
35
36
|7
38
39
Nom.
1 2
A88 8317,843918541
• 9436954819660
589 o555 0667 0779
1674 1 786] 1 898
2792 2904 3oi6
34
8653 8765
39104022
5o28 5i
61456257
7263 7374
83798491
94969608
0724
i84o
2956
395914071
590 061 2
1728
2844
5075
6189
73o4
84i8
• 9532
591 0646
1760
2873
0757
1871
3984
39864097
5098
6210
7322
8434
• 9546
592 0657
1768
2878
3988
5098
6208
73i8
84^7
5936644
1753
2861
3968
5076
61 83
4i34
4o4a5i
6369
7486
86o3
5i86
63oi
74i5
853o
96449755
5209
6322
7434
8545
9657
0768
1879
29893
40994
5209
63i9
7429
8538
9536 9647
0755
1 863
7290
8397
• 95o3
5940609
I7i5
2820
2971
407
5187
6294
74oi
85o7
9614
0720
1825
2931
• 9447;9^58
595055 1, 066 1
1654,1764
275712867
386o
977a
0891
2009
3i2d
9884
ioo3
2121
3239
4246 4357
36 004
08'
1951
3*167
4i83
529854095521
6412 65246635
7527 7638 775o
864187538864
08690980
1982
3095
4208
5321
6433
7545
8656
9768
7656
8768
9879
08790990
1990
100
210
5320
643o
94
8649
975?
0866
1974
3082
190
5297
64o4
75ii
8618
9724
o83o
1936
3o4i
4257
5362
39264036 4 1 47
5o3o'5i4i 5-^5
61 35'6246 6356 6466
7239 735o 7460
8344,8454 8564
0772
1875
3978
5363
648i
7598
8714
I
6947
ao63
3179
4294
98669978
2093
3207
4319
5432
65446655
2101
3211
4321
543 1
654 1
7540 7650
8760
9868
'>977
2085
3193
43oi
54o8
65i5
7622
9835
0941
2046
3[53
5475
6593
7700
8826
9943
1059
2175
3290
4406
1091
2-2o5
33i8
443 1
5543
7767
8879
9990
not
3212
3322
4433
5542
6652
7761
8870
9970
1088
2196
33o4
44ii
55i9
6626
7733
8729I8839
9945
io5i
2i57
3262
4368
5473
6577
7571 7681
8675 8785
966897789889
0882099:
1985
3o88
4191
2
2695
3198
43or
5 6
88778980191
9996 ♦10802
iii5
2233
335i
4469458
5587"'
5698
67046816
7933
9049
7821
8938
♦o54 0166 0277
1 170
2286
3402
45i7,
5632
6747
7861
5744 5855
6747I68586970
" 79738084
9198
0201 o3i2
89759087
♦089
1203
33i6
3439
4542
5654 5765
6766687869897
7878
899091
♦ lOI
1212
2323
3433
5653
8981
♦0900201
1 1 99
23o7
341 5
4522
563o
6737
7843
♦o56
1 162
2268
3373
4478
5583
6687
7792
1227
2345
3463
9
9325
0443
i45o i562
25692680
36873798
4693148044916
5810IS9226034
7i5i
8268
7 8
01 9213
6220I0332
1 338
3457
3575
6927
80448
1394
25o9
3625
46294740
1282
2398
35i3
i3i4
3437
3540
4653
7989
01
K>2I2
i323
3434
3544
4544 4655
5764
6763 6874 6985
787a 7983
9092
1 3*^9
3417
3525
4633
5740
6847
7954
89509060
999?
iio3
2206
33o8
44ii
0167
1273
2378
3483
4588
5693
7902
9161
5392650
1426
3652
4764
5877
8101
9212
o323
1434
2545
3655
5875
8094
9203
o3i2
1430
2528
3636
4744
585 1
6958
8o65
9171
0277
1 383
3489
3594
58o4
67986908
8012
S922
7o3o
8i56
9273 9384
0389
i5o5
3621
3736 3848
4852
7681
8196
9310
u433
1537
265o
3763
100
8212
o5oi
1617
2732
4963
59676078
7193
83o7
9421
o535
i648
2761
3874
48764987
59886099
721 1
8323
9333,9434
0434
1 545
2656
3766
47664876
59866697
70967307
8205
9314
0423
i53i
2639
3747
4854
5962
7069
8175
9282
o388
1494
2599
3704
469948094920
5914
7010
8123
8895 9006 91 16 9227
o33o
r434
2537
0220
i323
2426
♦110
12l3
23i6
3419352913639
4521 "
4633
4742
o546
i656
2767
3877
4987
83 16
9425
o533
i64a
2750
3858
4965
6072
7179
8286
9392
0498
1604
2710
38i5
6625
7129
8233
9337
o44i
i544
a647
3750
4852
OiC
3970I4080
I 2 3 4 5~6 7 8 9 I
I
a
3
4
5
6
7
8
9
Its
224
316
448
67a
784
1008
III
Diir.
I
III
a
222
3
333
4
444
5
555
6
665
I
777
888
9
999
I
2
3
4
5
6
7
8
no
a2o
33o
440
55o
660
770
880
95^
Diftltllalt.
LOGARITHMS OP MUMBBBS.
53
3MO
4i
4a
43
44
45
46
47
48
49
30 50
5i
52
53
54
55
56
5?
58
59
3960
61
63
63
64
65
66
67
68
69
3970
71
7a
73
74
75
76
77
78
79
3980
81
83
83
84
85
86
87
88
89
3990
9»
93
94
99
ftW496a
6064
716673767387
8368 8378 8488 8598 8708
• 93699479958996999810
596 0470 o58o 0690 0800 0910
1571
3671
3771
4871
5971
7070
8160
• 9368
59703670476058606960806
i465
366i
4758
563 3673
5855 5965 6074 6i84 6294
6953
8o48
• 9145
5980341
i336|
3433
3537
4633
5717
6811
7905
^ «999
6648
883i
• 99"
6ooioi3
3io3
3193
601 0817
1905
6355
7341
8438
• 95i
5073
6175 6385 6395 65o5
1681
3781
388i
4981
6081
71
8379
9378
8073
1 575
3673
3770
4868
3770 388o
4977
7063
8i58
9354
o35olo46o|6
i446|
a54i
3637
473 1
58a6
6930
8014
8999[9io8
o3oa
1395
3388
^V9
3371
4464
5556 5665
3481
4573
6757
7739 7849 7958
♦o3i
iiaa
aaia
33o3
4383 4393 45oi
5373 5483 5591
646a 657 1 6680
755 1 76607769
864087498858
9739 9838 9947
0926
aoi4
3l03
41
894
5i68 5377
5385
6363647a
745o 7559
85378645
4I96339731
2 8^
5i83 5293
5 6 7 8 9
1664
3783
7171
8368
9364
i556
265 1
3746
484i
59366045
7630
8124
9318
o3ii
i4o4
a497
3590
468a
5774
6866
894090499158
io35
21 a3
32II
398
7497
1901
3ooi
4ioi
530I
63oi64ii
74007510
^
1794
3892
7281
8377
94749583
0671
177
569
665
166
2761
3856
4950
71397249
8233
9327
4791
5884
6975
oi4o|o349
I23l
a3ai
34ii
1 340
243o
3520
♦o55
ii44
2232
33 1
4407
5494
658i
7667
8754
9840
54o3
7607
201 1
3iii
4211
53ii
8609
9708
1904
3oo2
4099
5197
7301
8487
2870
3965
5o6o
61 54
8343
9436
0420 o53o
i5i4 1623
36063716
3699I3808
4901
5903
7064
86678176
9367
o358
1449
2539
3639
89679676
or64
1253
2340
9i34a8
45i6
56o3
6690
7776
8663
9948
55i3
66i5
7717
8818
9930
1030
3I2I
3221
4321
5431
6531
7630
871
9817
091
988
♦637
6ti636|ii35
4a
«34
3221
4319
54i6
64o3|65i3
750076
201
3ll2
4209
53o6
8597
9603
0789
1884
9803089
6754184
5169 5279
6a64637S
6a64
7358
8452
0639
1732
2825
3918
5oio
7194
8285
0467
1 558
2648
3738
4828
5918
7007
8096
9185
0273
i36i
a449
3537
4624
5711
^98
7885
8971
♦057
5624
672568366946
7827 7937
8939 9639 9149 9259
♦o36
ii3i
223l
333 1
443i
553 1
66316741
773o
29
995»7
10
8706
960a
0898
1994
637
7467
656i
95469655
0748
i84i
3934
4027
5119
6102 621 1
73o3
8394
93769485
0576
1667
2757
3847
4937
6627
7116
8205
9294
o382
1470
2558
3646
4733
5820
6907
5734
01 4o
1 241
2341
3441
4541
564i
7840
89399648
7993
9680
016616274
2233
333 1
4439
5536
6623
7719
8816
9912
1008
2io3
3i99
4394
5388
6483
7577
8671
9764
o858
1951
3o44
4i36
5aa8
6320
74i3
85o3
95q4
0685
1776
2866
3956
5o46
6i36
7225
83i4
9402
0491
1 579
2667
3754
484J
5920
7016
8102
9188
5844
8047 8] 58
0256
i35i
245 1
355i
465 1
5751
685o
7950
oi47
1345
2343
3441
4538
5636
6733
oi3i
1227
2322
3418
45i3
5607
6702
7796
9874
0967
ao6o
3i53
4245
5338
7521
8612
9704
0794
i885
2975
4o65
5i55
6244
7334
8423
951 1
0600
1688
3776
3863
4950
60376
7ia4
8211
9297
o383
5954
7o56
o36o
i46i
256i
366t
4761
586i
6960
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9i56
0237
1 355
2453
355i
4648
5745
6843
783
892
♦022
1117
22l3
33o8
44o3
5498
6592
7686
8780I8890
9963
1076
2169
3262
4355
5447
64296539
7636
8722
9813
0903
1994
3o84
4174
5264
6353
7443
853 1
9620
0708
17^
2(
3972
5o59
146
7233
83 19
94o5
0491
012 34 56789
Diir.
no
T09
DifR
Diff. At MoU.
1
no
2
3
4
5
6
220
33o
440
55o
660
7
8
770
880
9
990
I
3
J?2
3
337
4
436
5
545
6
654
7
763
8
87a
9
981
I
3
3
4
5
6
7
8
108
ai6
334
432
540
648
756
864
9 97a
DUr. k. Molt.
B'
54
LOGARITHMS OF HUMBBSS.
NttlB.
4000
oi
02
o3
o4
o5
06
07
c>8
09
4o 10
II
13
i3
i4
i5
16
n
18
»9
40 30
ai
aa
a3
a4
a5
26
27
a8
29
40 3o
3i
3a
33
34
35
37
38
39
4o4o
4i
43
43
44
45
46
47
48
49
40 5o
5i
5a
53
54
55
56
57
58
59
Num.
12 3 4
5 6 7 8 9
O02o6oo;o7o8
1686
a77i
3856
4941
6oa5
7109
2879
39<i4
5o49
0817,(3936
1794 1903 2011
2988
4073
5i58
6134624a 635i
3096
4181
5266
• 9277I9385
6o3 o36i o4()9
1444 1 55?
2527 a635
36093717
4693 48cK)
5774 588 a
7aib|7326
6855
7937
9018
604 0099 020-7
Ii8i> 1288
8410
9494
o577
1660
3743
3826
4908
5990
6964 7072
8045 8i53
9335;
913b
2361
33ii
443 1
55cx)
658o
7435
85i9
960J
0686
1769
285 1
3934
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2120
32o5
4390
5375
645o
7543
8637
9711
0794
1877
3960
4o4a
5oi6l5ia4
60986206
7180
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3477
3557
4637
5716
7659 776-»'7875 7983
7388
8370
945i
o53a
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i5o4
3585
3665
4745
5824
690317611
8735
• 9816
6050895
1973
3o5o
4r28
52o5
6283
7359
8435
• 9513
606 o5H7
i6()3
2739
38i4
4889
5963
7<>37
8111
• 9185
6070359
i332
34o5
3478
455o
5633
6604
7766
8837
A 9909
0000979
ao5o
3l20
4 1 91
8846
9994
1002
2080
3i58
4236
53i3
63906498
8953
•o33
1 1 10
2188
3366
4343
5431
7574
865 1
5i
7467
8543
9619
0695
1771
3846
3921
4996
6071
7145
8219
9393
0366
1439
35l3
3585
4657
5730
6803
7873
8945 9052
9061
of4o
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2296
3374
445 1
5528
()6o5
7682
8758
9737 9834
08030910
1878IT986
39541 3o6i
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OJ 521 I
6385
6178
7353
8336
94<x)
0473
1 547
3630
3693
4765
5837
6909
7980
7360
8434
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i654
3737
3482
4559
5636
6713
7790
8866
994a
1018
joo3
3169
4344
53i8
6393
7467
8541
9615
0688
1761
3834
♦016
1087
3 1 57
3337
4398
OI23
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3264
38oo'3907
48724979
59446051
7016J7123
80878105
91599266
02 3o 0337
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337113478
33343441.3548
44o4 45iil46i8
ii43ii25i
aa28!3337
33i3
4398
5483
6567
765 1
8735
9819
0902
1985
3o68
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5233 5341
63i5
3433
4507
5591
6676
7760
8844
99^7
1010
3093
3176
4359
7396
8478
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0640
1720
2801
388 1
4961
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7119
8198
9377
o355
1434
a5ia
64a3
75o4
8586
0748
1838
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5
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83o6
9385
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1 541
3619
35893697
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5744 585 1
683 1 6938
7897
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1135
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3276
435 1
5426
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7574
8648
9732
0705
1869
3941
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0444
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a585
3656
4735
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3445
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1119
3203
3384
4367
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653 1
7&i3
8694
9775
o856
1936
66396747
017
4097
5176
6a56
7335
84i4
9493
0571
1649
2737
38o5
488a
5959
7o36
8oo58iia
9081
oi57
1333
33o8
3384
445o
5533
6608
7683
8756
9820
0903
1976
3o49
4iar
5o86 5i94 53oi
6i58 6366 6373
733o 7337 7445
83oa84o985i6
9373 9480 9587
o55i
163a
369a
3763
4833
9180
0265
1 340
34i6
3491
4566
564i
6715
7789
8863
9937
1010
ao83
3i56
4aa9
1 577
a66a
3747
483a
59x7
7001
79778085
90609169
i468
a554
3639
4734
58o8
0144
iaa7
a3io
3393
4475
5557
7721
880a
9883
0964
ao44
3ia5
4ao5
5384
6364
7443
R5aa
9601
0679
1757
3835
391a
6007
7144
8aao
4674
5748
68a3
7897
8971
♦044
1117
31
32
4336
54o8
648o
oa5a
1 335
a4i8
35oi
4583
5665
7829
8910
999<
107a
3i5a
3a33
43i3
539a
6473
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9708
0787
1 865
3943
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5098
6175
7a5i
83a8
929619404
037a
1 448
3533
3599
o48o
1 556
263i
3706
4781
5856
6930
8004
9078
01 5i
iaa5
3398
3371
4443
55i5
6587
o658
1739
3870
4939
7553 7659
8623 8730
969419801
0765
i836
2906
3977
5o46
0872
1943
3oi3
4o84
5i53
01234 56789
IMC
108
107
Diir.
DUCi
X
a
iri
3
3*7
4
436
5
545
6
654
7
763
8
87a
9
981
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fo8
a
a 16
3
3a4
4
43a
5
540
6
648
7
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8
864
9
97a
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107
a
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3
3ax
4
4^
5
535
6
64a
7
8
'^
9
o<B
Diir.<
/
L001.EITHM8 OF NUMBERS.
55
1 2
406O
6i
6)
63
64
65
66
67
66
69
4070
71
72
73
74
75
76
77
TB
79
4o8o
81
8a
83
84
85
86
«7
88
89
4090
91
9®
99
4100
01
01
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o4
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06
07
08
09
4i 10
II
12
i3
M
i5
16
17
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6171
7233
8295
• 9356
6120417
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2
5720
• 99^7
6i3 ioi5
2074
3i32
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63o4
736 1
84i8
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6i4o53i
1 587
2643
580959
6863
7918
5474
6544
7613
8682
96449751
608526o5367
633o6437
73907506
84688575
« 9537
60906050712
1674 1 78 1
2742 2849
38093916
48774984
5944 6o5i
7011 71 18
8078 8184
• 9144 9251
610 0210 o3i7
1276 i383
23422448
34073514
4472
5537 5644
66026708
76667772
8730 8836
• 97949900
61 1 0857 0964
1921
2984
4o46
5io9|5
4579 4685
2027
3090
4i53
2l5
6277
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9462
0524
1 584
5392645
3706
4766
677968856991
7944
9004
♦062
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295
5353
64io
7467
6524
9580
0637
1693
2748
38o4
4859
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4o23
5090
6157
7224
8291
9357
0423
3620
4685
5750
68i5
7870
8943
♦007
1070
2i33
3196
4259
5321
6384
7445
85o7
9569
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1691
2751
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4872
582615931
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1227
1121
217912285
3343
44oi
5459
65i6
7573
863o
9686
0742
1798
2854
3909
4965
6020
7074
3
3587
665i
7720
8780
9858
5 6 7 8 9
4i3o
5197
6264
733 1
8398
9464
o53o
1596
2661
3727
479a
5856
6921
7985
9040
oii3
1 176
2240
33o2
4365
5428
64906596
7552
861 3
9675
0736
1797
2857
3918
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8o5ol8i56
9215
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5070
6125
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8129(8234
5688
6758
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7027
8092
91 56
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1439
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9897
0954
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2208
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4343
54ii
6478
7544
8611
9^7
0743
1809
2874
3940
5oo5
6069
7134
8198
9262
6326
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2452
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4578
564o
6702
7764
8826
9887
0948
2009
3069
4i3o
5190
6249
7309
8368
9427
0486
1 544
2603
366i
4718
5776
6833
7890
8947
♦oo3
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21l5
3171
4226
5281
6336
5902
6972
8o4i
91 10
0178
1246
23i5
3382
4450
55i7
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765 1
8718
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1916
2981
4o46
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6176
7240
83o4
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0432
1495
2558
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6808
7870
8932
3
io!>4
2ll5
3175
4236
5396
6355
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2708
3766
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588 1
6939
7996
9052
0109
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2321
3276
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6442
7391 7496
8445l855o
6000
7078
8i48
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1 353
2421
3489
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5624
6691
7758
8824
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6282
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3728
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5852
6915
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♦099
1 160
3221
3281
4342
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16116
7185
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9328
0392
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2528
3
4
5731
6798
7864
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2129
3104
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5324
2614
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91 58
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2326
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9144
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7627
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8207
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2433
3487
4543
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7707
8761
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7292
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1 567
2635
3703
4770
5837
6904
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♦ io4
1170
2235
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4366
543 1
7560
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8867
01234 56789
Diff.
106
Diir.
DiiT. k, Mnh.
I
107
2
214
3
321
4
428
5
535
6
642
7
8
749
856
9
963
I
fo6
2
212
3
3i8
4
424
5
53o
6
636
7
742
8
848
9
954
I
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2
210
3
3i5
4
420
5
525
6
63o
7
735
8
840
9
945
Dim k. Muh.
56
LOGARITHMS OP NVMBXRS.
Nam.
4120
21
aa
a3
24
a5
a6
aS
a9
4i3o
3i
3a
33
34
35
36
3?
38
39
4i4o
4i
43
43
44
45
46
47
48
49
4x5o
5i
5a
53
54
55
56
5?
58
59
4i6o
6i
63
63
64
65
66
67
68
69
4170
71
72
73
74
75
76
77
78
79
Nam.
12 3 4
614 8973*9078 9183938819394
61 5 0036
1080
ai33
3187
4a4o
5393
6345
7397
8449
• 9501
6i6o552
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a654
3705
4755
58o5
6855
7905
8954
6170003
io53
aioi
3i49
4197
5345
6293
7340
8387
• 9434
618 0481
i527
a573
36t9
4665
5710
6755
780CJ
8845
• 9889
6190933
1977
3o2r
01 33 0237
ii85 1 391
2339 3344
3393 3397
4345 445o
5397; 55o3
645o6555
7503I7607
8554.86598764
9816
0867
1918
3969
4030
96069711
o657 0763
1708 i8i3
3759 3864
38io39i5
48604965
59io|6oi5
6960I7065
8oioi8ii5
90599164
oio8i03i3
1362
ll5"7
22o6
3254
43o2
535o
6398
7445
8492
9539
o5S6
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2678
3724 3828
23ll
3359
4407
5455
65o3
755o|
8597
9644
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17.J7
2783
4769
58i5
6860
7905
8949
9994
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2082
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61 5o'
7193
8335
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6254
7297
8339
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6009
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♦09S
1 143
3186
332
427
53i6
1
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7401
8443
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3402
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5534 5638,5733
1 465' 1 569
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61
3063
7170
8330
93699374
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1 367
241 5
3464
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6607
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2887
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4979
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7069
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• 9^4
631 0734
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0828 0933
1346
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35
7275
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0433
1473
2520
3569
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5664
6713
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8806
9853
0900
1946
2903
4o38
5o83
61 38
7173
8218
9363
o3o7
1 35!
23(^5
343S
448 1
5524
6567
7610
8653
0736
1777
2818
3859
4900
5941
6981
8021
9061
♦ 100
1 139
1607
a66o
3713
4766
58i8
6871
7933
897590809185
♦oa6oi3i
1078 1 1 83
ai
3i79
4a3o
393334
3a84
4335
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633o6435
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9479 9584
o5a8
1 577
2635
3673
4721
5769
6817
-7864
891 1
9958
ioo4
ao5o
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wnw 01234 56789
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DMT. Jt M sk.
60
LOGARITHMS Of NUMBERS.
Suau
4360
6i
6a
63
64
65
66
67
66
69
4370
71
72
73
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75
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78
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81
83
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Num.
12 8 4
630 4865
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DiAltMalt.
LOOARITHaif OF NUMBERS.
61
N... 12 3 4
4430
21
23
23
24
25
36
27
28
^9
44 3o
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32
33
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37
38
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2683
Dir.
ir«a 01234 567891
98
97
Vitti
Diff. k. Huh.
I
2
3
4
5
6
7
8
I
2
3
4
5
6
7
8
9
I
2
3
4
5
6
7
8
196
394
393
490
588
686
784
883
194
391
388
485
583
873
96
192
288
384
480
576
67a
768
864
DUr. 9t Malt
62
LOGARITHMS Ofe N U M B B B 8 .
N
4480
6i
89
83
84
85
86
87
88
89
4490
91
92
93
94
95
96
97
98
99
45 00
01
oa
o3
04
o5
06
07
08
09
45 xo
II
la
i3
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16
17
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45 20
ai
aa
a3
24
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27
a8
29
45 3o
3i
3a
33
34
35
36
37
38
39
Num.
12 3 4
•51 2780 2877 2974 3071, 3 168
3749 3846 3943 4o4oJ4i37
4719 481 5;49i 2 5oo9 5io6
5687 5784i588i
66566753;685o
5978
6947
76247721 7818 7915
8593869087868883
• 9561I9657
653 o528|o625
1496 1593
2463 2 56o
3431I3527
439714494
5364 5461
633 1 6427
653 01
9754
0722
1690
2657
36a4
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5558
6524
7297 7394 7490
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1160 1256
2125 2222
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6273 6369
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3980
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0224
1 188
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4076
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5904*6000
1 480
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4266 4362
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♦039
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3712
4633
5554
6475
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656i
7309749a
833o 8423
9354
0285
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2i46
3076
4oo6
4935
6794
7723
865a
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1 636
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6427
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8272
9195
0117
1039
1961
2882
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5646
6567
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923619328
Nii. 12 3 4
5 6 7 8 9
93
I
a
3
4
5
6
7
8
I
a
3
4
5
6
7
8
9
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ife
279
37a
465
558
65 1
744
837
184
276
368
460
55a
644
736
828
DiCASiiat.
F'
66
LOGARITHMS OF NUMBERS.
tLMmh I
NVBBa
4720
31
92
a3
95
26
27
38
39
'47 3o
3i
33
33
34
35
36
37
38
39
47 4o
4i
4a
43
44
45
46
47
49
47 5o
5i
53
53
54
55
56
57
58
59
4760
61
62
63
64
65
66
67
68
69
4770
71
73
73
74
75
76
77
78
79
Nun.
12 3 4
5 6 7 8 9
• 9430
674 o34o
1360
2179
3099
4018
4q37
58S
6775
7693
861 1
6756447
1 365
3383
56 S
9539 96
3300
4117
5o34
7783
8700
• 9615
6760531
1447
3363
3377
4193
5io7
603 3
6936
785o
8764
• 9678
677 0593
i5o5
3418
3333
4344
5i57
6070
7
8806
• 9718
678 0639
i54o
3453
3363
4273
5i84
6094
7004
7914
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1 552
346i
3370
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56
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4303
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3375
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2693 3784
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2907
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6549 664
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2916
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5489
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7336
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1916
3833
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9346 9438
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4o35
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01234 56789
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Diff.
I
3
3
4
5
6
7
8
9
I
3
3
4
5
6
7
8
9
276
368
46o
55a
644
736
838
189
973
364
455
546
%
819
Diff. * Mall.
w
LOeABITHMS OF NUMBERS.
67
4TO0
8i
83
83
84
85
86
»7
88
4790
9'
94
99
4800
01
oa
o3
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17
18
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27
28
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34
35
36
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38
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5187
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7912
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4262
5 168
6074
6980
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6934
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5963
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4i 59 4249 4339 4429 4520
5241
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181411904
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6 6 7 8 9
1722
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1642
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4234
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6928
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7614
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1 191
2092
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6oo5
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1 281
2182
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2977
3785 3875
4683
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7377
8274
4773
5671
569
7466
8364
1234 56789
90
Dur.
DiCJcMnh.
I
2
3
4
5
6
7
8
I
2
3
4
5
6
7
8
I
2
3
4
5
6
7
8
9
182
273
364
455
546
637
728
819
270
36o
45o
540
63o
720
810
178
267
356
445
534
6a3
712
801
Difll^JCvlt.
68
LOGARITHMS OF NUMBERS.
Nam.
4940
4i
4a
43
44
I
45
46
47
48
49
48 So
5i
52
53
54
55
56
57
58
59
48 6o
6i
6a
63
64
65
66
67
68
69
4870
71
7a
73
74
75
76
77
78
79
4880
81
8a
83
84
85
86
. 87
88
89
4890
9»
93
94
99
Num.
684 8454
• 9351
685 0248
1 145
ao4f
2933
3834
473o
5626
6522
7417
83i3
• 9208
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1721
2613
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5290
6181
7073
7964
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6880637
1528
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3089
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8633 87238813
9530 96209710
0427 o5 1 7 0607
1 324 Mi4 i5o3
2221 2810 2400
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6791 6880
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3o27
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5716
6611
75o7
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9297
0192
1087
1962
2876
3770
4665
5558
6452
7346
8239
9182
♦025
0018
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2703
3595
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5379
6270
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5 6 7 8 9
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1 177
2071
2
3
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1900
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2608
35o2
4396
5290
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9261
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9118
♦01 3
0908
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2697
3592
4466
538o
6273
70787167
79718060
8864 8953
9757
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2435
3327
4219
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6oo3
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7786
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1 349
a 240
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4020
T
6689677
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8468
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291 1
4687
5575
6462
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8a37
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♦010
0807
1784
9846
0739
i63a
2524
3416
4309
5200
9446
o335
1223
2II2
3ooo
12 3 4
5 6 7 8 9
Dur.
«9
DiC^tMnh.
I
a
3
4
5
6
7
8
I
a
3
4
5
6
7
8
9
180
270
36o
45o
540
63o
720
810
.5
267
356
445
534
6a3
712
801
DUr. I DiflllbMult.
LOGARITHMS OF NUMBERS.
69
Num.
4900
oi
02
o3
04
o5
06
07
08
4910
If
12
i3
14
i5
16
17
18
'9
4920
21
22
23
M
25
26
27
28
29
4930
3i
32
33
34
35
36
37
38
39
4940
4i
42
43
44
45
47
48
49
4950
5i
52
53
54
55
56
57
58
59
If HID.
12 3 4
5 6 7 8 9
6901961
2847
3733
4619
55o5
6390 6479
7364
8249
91 34
♦019
7275
8161
9046
• 9930
6910815
1699
2584
3468
4352
5235
6119
7002
7885
8768
• 9651
692 o534
i4i6
2298
3i8o
4062
4q44
5826
6707
7588
8469
• 9350
6930231
nil
1991
2872
3753
463 1
55ii
6390
7269
8149
9027
• 9906
6940785
i663
254i
3419
4297
5175
6o52
6929
7806
8683
• 9560
6950437
i3i3
2189
3o65
3941
2o4^
2936
3832
4708
5593
0903
1788
3672
3556
4440
5334
6207
7090
7974
8857
2i38
3o24
3910
47 '
2327
3ii3
23i5
3201
4087
4973
5770J5859
6567
7452
8338
9223
0107
1876
2760
3644
4528
5412
6295
7179
8062
8945
66566744
7541 17630
8426 85x5
9739 9828
0622
i5o4
2387
3269
4i5i
5o33
5914
6795
7676
8557
9438
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1199
2079
2960
3839
4719
64^
735-7
8236
9115
9994
0872
i75i
2629
35o7
4385
5262
6r4o
7017
7894
8771
9648
0710
1593
2475
3357
4a39
5l20
6003
6883
7764
8645
9526
0407
1287
2167
3o48
3027
4607
5687
6566
7445
8324
9203
♦083
0960
1839
2717
3595
4472
535o
6227
7io5
7983
8859
9735
4327
5309
609061
6971
7853
052406120700
i4oi 14881576
2277 2364 2452
3i 53 32403328
4o29l4ii6i43o4
93ii
0196
1080
ro65
2849
3733
4617
55oo
6384
7267
8i5o
9033
9916
0798
1681
3563
3445
9399
0284
1 160
2o53
2037
3821
47o5
5589
647a
7355
8238
9121
♦oo4
0887
1769
365 1
3533
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5297
6178
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7941
8733 8822
96149702
0495
1 375
2256
3i36
40 1 5
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5775 5863
6654
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6742
7621
84i2 85oo
9391
0170
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3805
3683
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5438
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7193
8069
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937
035
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20l4
3892
3770
4648
5526
64o3
7380
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9034
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0787
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2540
3416
4291
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32901 3379' 3467
4176 4265
5062
5947
6833
7718
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6o36
6921
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86038693
9576
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3ii4
9488
0373
1257
2l4l
3026
3910
4793
5677
656o
7444
8327
9210
0092
0975
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2739
3621
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6266
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8910
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067 1 10759
5765
6649
7533
84i5
9398
0181
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1945
2828
3710
4591
5473
6354
7236
8117
8098
9878
i55r
2432
33x3
4191
5071
5951
683o
1639
2520
3400
4279
5i59
6039
6918
7709 7-^97
8588
9467
o345
1224
2103
2980
3658
4736
56x3
6491
7368
8245
9x22
9998
0875
I75i
2627
35o3
4379
8676
9555
0433
l3l3
2190
3o68
3946
4834
5701
6578
7456
8333
9209
♦086
0062
i83q
2715
3591
4467
4353
5239
6124
7010
7895
8780
9665
o55o
1434
23i8
3203
4086
4970
5854
6737
7620
85o3
9386
2670' 3758
3556 3644
02
u5i
3o34
2916
3798
4680
556 1
6443
7324
8205
9086
9967
0847
1737
3608
3488
4367
5347i
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5337
62x3
7098
7984
88
975
0638
6989
5398-
1 522
2407
3291
4175
5o58
5942
6825
7709
8593
9474
690357
x24o
2122
3oo4
3886
4768
5649
653 1
74x2
8293
9174
♦o55
0935
i8i5
2696
3576
4455
5335
453o
54x6
63o2
7187
8072
8957
42
0726
1611
2495
3370
4263
5i47
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6914
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61 36162 1 4
70067094
7885 7973
876418853
9643 9730
o5ai
1399
3278
3x56
4o34
4911
5789
0609 0697
1487
2366
3344
4X31
4999
5877
66666754
7543 763 1
843o85o8
9397 9385
0174
io5o
1936
3802
026X
1x38
20l4
3890
36783766
4554'4642
9563
0445
1 328
2210
3092
3974
4856
5737
6619
75oo
838i
9262
0x43
X033
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3784
3664
4543
5423
63o2
7182
8061
8o4o
9818
1575
2453
333i
4209
5087
5964
6842
77x9
8596
9472
0349
X225
2X02
2978
3854
4729
01234 56789
DUr. Dim^tMoh.
88
Diir.
I
2
3
4
5
6
7
8
X
a
3
4
5
6
7
8
I
2
3
4
5
6
7
8
178
267
356
445
534
623
71a
801
88
176
264
35a
440
528
616
704
79a
Diff. it Mult.
70
LOGARITHMS OF N U IC B B R 8 .
Nun.
4960
6f
6i
63
64
65
66
67
68
69
4970
71
7a
73
74
75
76
77
78
79
4980
81
8a
83
84
85
86
87
88
89
4990
9»
93
94
96
98
99
5ooo
01
oa
o3
o4
o5
06
07
08
09
So 10
II
ra
i3
i4
i5
16
17
18
19
Num.
12 3 4
8
9
SSBT
|648o
7355
8a3o
Dir
DULtLibtL
695 481714904 4992 5o7Q
5692
6568
7443
83i8
9193
69600(77
0942
idi6
3690
3564
4438
53m
6i85
7o58
7031
88o4
• 9676
6970549
1421
22
295
3i65
4o37
4909
5780
6652
7523
8304
• 9264
6980135
5780! 58671 5955
6743 683o
6655
753o
84o5
92S0
oi55
loao
1903
2777
365 1
4525
5399
0372
71457233
7618
8493
9367
oa4a
II 16
a865
3739
461 a
5486
6359
8018
8891
9764
o636
i5o8
8io5
8976
985 1
0723
77o5
858o
9455
o33o
iao4
2078
2952
3826
4700
5573
6447
73ao
8193
9066
9938
()8ii
1596 i683
238 1 9468
3353;334o
4l34!42I2
49965083
5M7
6739
7610
848 1
9352
0232
5955
6826
71
85i
697
568
9439 9536
03090396
2555
3427
4299
5170
6o4a
6913
7784
8655
6042
6918
7703
8668
9542
0417
2166
3o4o
3913
4787
566i
6534
7407
8280
9153
♦oa5
0898
1770
264a
35i4
4386
5257
6129
7000
7871
874a
9613
o483
963o
o5o4
1379
2253
3i27
4ooi
4874
5748
6621
7494
8367
9240
oii3
0985
1 857
272
36oi
4473
5345
6a 16
92817
7087
7q58
8§
9700
0570
7174 7361
8o45,8i32
a9|89i6|9oo3
978Tip874
06570744
i44i
a3ii
3i8r
4o5i
4920
5790
6659
7528
8307
9266
01 34
ioo3
1871
2739
3607
4475
5342
6210
7077
7944
8811
9677
o544
i4io
2276
3i42
4oo8
4874
5730
66o5
63o5
7180
8o55
9717
0592
1406
2340
32i4
4088
4962
5835
49
23
67096796
7582
8455
9327
107a
1945
456o
543a
63o3
8930 9018 9105
06790
1 554
a4a8
33o2
4i
5o4
59
764263
7669
8542
94i5
02ootoa87
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2904
36893776
4647
5519
6390
i5a8
3398
3368
4i38
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i6i5
3485
3355
4224
5094
58775964
67466833
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84848571
9353
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725o
8II7
669116778
557^
6393
7268
8143
98029080
07070854
i64i
25i5
3389 3477
5i37
435o
5224
60106097
6883
7756
6070
7644
86298716
95oa
o374
1247
2119
170a
2572
3442
43ii
5i8i
6920
7780
8658
9439 9526 96
0395
1263
2l3l
^7
4735
56o2
6470
7337
8204
9071
9937
0804
1670
2536
34oa
4268
5i33
54
1728
a6o3
9589
0462
1 334
2206
3078
3950
4735 482a
56065693
6477 6565
73497436
8220 83o7
9177
♦048
0918
1789
2659
3529
43o8
5268
6o5o6i37
7007
7876
8744
i3
0482
i35o
3218
3o86
3954
4822
5689
6556
7424
8291
9157
♦024
0890
1757
2623
3489
4354
5230
6o85
695
12 3 4
M-
»7
I
2
3
4
5
6
7
8
9
88
176
264
352
440
528
616
704
79a
I
9
3
4
5
6
I
6 7 8 9 I iMit DiCfcMvk-
L
LOGARITHMS OF M0MBESS.
71
5090
ai
ai
a3
a4
35
a6
a?
»9
5o3o
3i
3a
33
34
35
36
3?
38
39
5o4o
4i
43
43
44
45
45
4r?
48
49
5o5o
5i
5a
53
54
55
56
5?
58
59
5o6o
6i
6a
63
64
65
66
67
68
69
5070
71
7a
73
74
75
76
77
76
79
01234 56789
TOO 7037
7902
8767
• 963a
70I
7xa4|7aio
8075
8o4o
9805
8^
9718
0496 o583 0669 07
43o5
5167
6oa8
77
86ia
• 947a
7o3o333
1 193
ao54
3914
3774
4633
635a
7313
8071
• 97W
704 0647
i5o5
3363
3aai
4079
4937
5794
665a
75oo
8366
1793
3649
35o5
436o
5a 16
607a
6937
778a
8442
93o5
0167
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189a
3754
353o{36i6
4989I5075
5853
6716
7579 7665
" 8528
9391
4392
5a53
6ii5
95
o4
1379
ai4o
4478
5339
6aoi
7062
79338009
87848870
386o
4719
557<
7ao8
8157
89309015
9»74
0733
1591
33o7
4i65
5oa3
588o
16738
7595
845a
705 0080^0 i65|oa57
0936
I033
1878
3734
3590
53oa
61 57
7383
8248
91 13
98019078
07560843
v534
2398
3262
4i25
9645
9o5o5
i366
2226
3ooo3o86
3q46
48o5
5665
16524
7383
9101
1677
2449] a535
3393
425i
5io8
33
7680
8537
9333930993949480
1107
iq64
3830
3676
44464533
5387
6343
70137098
7868I7953
7297
8i6a
9oa7
1630
3484
3348
4313
0354
III6
1078
3840
3703
4564
5425
6287
7148
973 1
0591
1452
23t2
3172
4o32
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8242I8328
9187
♦046
0904
7
8623
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1193
2049
2905
3761
4617
5473
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7184
8039
1706
2570
3434
4298
5162
6025
6888
7752
8614
9477
o34o
1202
2064
2026
3788
465o
55i2
6373
7234
8095
8o56
9817
0677
1 538
2398
3258
4ii8
4q77
5837
66106696
7555
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9273
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0990
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2707
3565
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1763
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0929
1703
2657
3521
4385
5348
611
6975
7838
8701
9563
0436
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3874
4736
5598
6459
7320
8181
9042
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0763
1624
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4304
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9286 937 a
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08490935
1710
2570
3430
4290
5 149
6009
6868
7727
8586
9445
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1 162
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2878
3736
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5453
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7166
8023
8880
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3162
4018
4874
5729
6585
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8295
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2830
5421
6284
7147
8010
8873
0598
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2323
3i85
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4909
5770
663 1
749a
8353
772917816
85q4I868i
1188
2o5a
2916
3780
4644
5507
6371
7234
8097
22
o685
1 547
2409
3271
4i33
9545
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1274
2i38
3002
3866
473o
5594
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9046
9908
0771
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2495
3357
4219
6717
5 5o8i
6 5942
6804
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43-76
5235
6095
6954
7813
8672
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0389
1348
2106
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5537
6305
7a52
8109
8966
9823
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1 536
2392
3248
4io4
^
6670
7526
838 1
7579 7665
844o 8526
9300
0161
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2743
36o2
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53a I
6181
7040
890798
758 884
875
9617
0475
1 334
2192
3o^f>
3908
4765
5623
648o
7338
8195
9062
9908
0765
1621
2477
3333
41
5o45
5901
6756
761 1
8466
9386
0247
1107
1968
2828
3688
4547
5407
6266
7126
7985
44
9702
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1419
2278
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5709
6566
7433
8280
9137
9994
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1707
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341
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t
5i3i
5986
684i
7697
8552
Vwm. 01234 56789
Dur.
86
Diff. lb Huh.
Diir.
I
2
3
4
5
6
7
8
I
2
3
4
5
6
7
8
9
I
2
3
4
5
6
7
8
86
172
258
344
43o
5i6
602
688
774
85
170
255
340
425
5io
595
680
765
Difr^Mnft.
72
LOGARITHMS OP NUMBERS.
Num.
5080
12 3 4
5 6 7 8 9
DtiT.
Diir.&Ma}L
705 8637
8723
8808
8694
%\
9065 ' 91 5o' 9336 9321J9406
6i
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9^77
9663 9748
99»9
♦oo5
0090 0176
0361
83
706 o347
0432
o5i8
o6o3
0688
0774
0859
0945
io3o
1116
63
1 201
1287
1372
1457
1 543
1628
1714
id??
i885
1070
3824
84
ao55
2l4l
3326
33l3
3397
3483
2568
3739
85
3910
2995
3o8o
3i66
3a5i
3337
3433
3507
3593
3676
86
3764
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4703
3934
4020
4io5
4190
4376
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4447
4533
87
4617
4788
4873
^^
5o44
5i3o
52i5
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5386
88
547'
5556 5642
5727
5898
6751
5983
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6325
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6837
6933
7007
7093
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86
5090
7178
7263 7348
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• 9737
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0675
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0845
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1 357
95
1442
1 537
1612
1698
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1 y
1783
1866
1953
38o5
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3039
3134
2309
688
96
3294
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2379
2465
3635
3730
3891
3076
3838
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9
774
97
3i4o
3232
33i7
3402
3487
3573
3658
3743
3913
98
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4o83
4169
4354
4339
4434
4509
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4680
4765
99
4935
5020
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5376
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5446
553i
56i7
5i 00
570a
5787
5873
5957
6043
61 38
63i3
6398
6383
6468
01
6553
6638
6724
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6894
783o
7064
7149
7234
7319
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74o5
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7575
7660
7745
7915
8000
8o85
8171
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834 1
8426
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8596
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8936
9033
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0078
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1 488
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1 574
07
1659
1744
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2?49
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2169
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3339
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08
a5o9
2594
2679
2764
3934
3019
3to4
3189I3374
09
3359
3444
3529
36i4
3699
3784
3869
3954
4039
4134
65
5i 10
4309
4394
4379
4464
4549
4634
4719
48o4
4889
4974
5833
II
5o59
5908
5i44
5329'53i4
53oo
5484
5569
64i8
565415739
la
6843
6078
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6333
6503J6588
6673
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6758
6938
701 3
7098
7i83
7268
7353
7437
7533
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7607
7692
7777
7862
7947
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8117
8202
8287 8371
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8456
854 1
8626
8711
8796
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8966
905 1
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1 1 73
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1427
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2021
3io6
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3375
336o
2445
253o
2615
5i ao
3700
2784
2869
2954
3o39
3i34
3209
3293
3378
3463
21
3548
3633
37.17
38o2
3887
3072
4820
4o57
4i4i
4336
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aa
4396
4481
4565
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4735
4904
4989 5c*74
583- 5933
5i59
nr
a3
5244
5328
54i3
5498
5583
5667
5752
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OD
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6091
6176
6261
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643o
65i5
6600
6684
6769
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3
3
170
255
a5
6939
7023
7108
7193
7378
7363
7447
7532
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7701
4
340
a6
7786
7871
7955
88o3
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8125
83IO
8294
837918464 8548
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425
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8887
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9057
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9236*931 1
9395
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9565
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7
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1089
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8
600
5i 3o
1174
1258
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1597
1682
1766
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1936
9
765
3i
2020
2io5
2189
2374
2359
32o5
2443
3528
26i3
2607
2782
3a
2866
2951
3<j36
3 1 30
3290
4x36
3374
3459 3543
3628
33
3713
3797
388a
3966
4o5t
4320
430^14389
4474
34
4559
4643
47^8
4812
4897
4982
5o66
5i5i
5235|532o
35
54o4
5489
6335
5574
5658
5743
5827
5912
^
6081^
6166
36
6a5n
6419
7365
65o4
6588
6673
6757
6927
701 T
37
7096
7180
7349
8195
7434
75i8
7603
7687
7773
7856
36
794'
8036
8110
8379
8364
8448
8533
8617
8703
39
Nna.
8786
8871
8955
9040
9134
9209
9293
9378
9463
9547
12 3 4
5
6
7
8
9
Diff.
Diff.lt Molt
LOGARITHMS OF NUMBERS.
73
Km.
5140
4i
42
43
44
45
46
47
48
49
5i5o
5i
5a
53
54
55
56
57
58
59
5i6o
6i
6a
63
64
65
66
67
68
69
5170
71
72
73
74
75
76
77
78
79
5i8o
8[
8a
83
84
85
86
87
88
89
5190
91
93
94
99
Nimn.
1234 56789
• 963i
711
9716
6476|o56i
i4o5
aa5o
3094
l32I
ai65
3oio
3938
478a
56a6
6470
73i3
8i57
9000
97599843
' 0686
i5a8
3854
4698
554a
6385
7339
807a
8915
71a 0601
1444
aa87 ^371
3ia9
3071
48i3
5655
7339
8180 8a64
90a I
• 986a
71 3 0703
1 544
a385
3aa5
4o65
4905
5745
6585
74a 5
8a64
9104
• 9943
714078a
i6ao
a459
32
4i
4q74
58ia
665o
7488
83a5
916a
7i5oooo
0837
1674
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3347
4i83
5019
5856
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8363
9198
3ai3
4o56
4808
5739
658i
74a3
0645
1490
a334
3178
4oa3
4867
5710
6554
7398
8a4i
9084
9927
0770
i6i3
a455
3a98
4i4o
498a
58a4
6665
7507
8348
9105
9946
0787
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3469
3309
41494
t
9187
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0866
1704
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338i
4220
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6734
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9a8a
98009885
91899374
♦o3i
0871
171a
a553
3393
3
5073
5913
66696753
7509
8348
75o3
84Ja
9271
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a6a7
3465
43o4
5i42
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346817
7655
84o3
93^
0167
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2678
35i4
435o
5187
6023
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9365
0729
1574
2419
3263
4107
4951
5795
6638
7482
8325
9168
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1697
2539
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675o|6834
i65o
25o3
3347
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9253
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1781
2624
3466
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01 15
0956
2637
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4317
5i57
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9355
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187a
371 1
3549
4387
5326
6o63
6901
7739
8576
9414
025l
1088
1925
2761
3598
4434
5270
6106
68596942
7778
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7^75
85i7
9358
0199
1040
1880
2721
356i
44oi
524i
6081
6921
7761
8600
943o
0278
1117
1956
2795
3633
4471
5309
6147
6985
7833
8660
9497
o335
1 171
aoo8
2845
368 1
45i8
5354
6190
7026
7861
8697
9449 953a
♦o54
0898
1743
2587
3432
4a
5l20
5q64
6807
765i
8494
9337
0180
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2708
355o
4392
5234
6076
764360
691
7759
8601
9443
0283
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1964
2805
3645
4485
5325
6i65
7oo5
7845
8684
9523
o362
1 201
2040
2878
3717
4555
5393
623
7069
7906
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9581
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1255
209a
a9a9
3765
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5438
6373
7109
7945
8780
8700a
oi38
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1827
2672
35i6
93697
35781
4445
5289
6i3a
6
7019
8663
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264 0349
iioi
2O04
2876
3719
456i
54o3
6245
7086
7928
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9610
045 1
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6o48
68
77^5
8578
9421
1107
1950
2792
3634
4477
5319
6160
7843
8685
9526
0367
1208
2048
2889
3729
4569
5409 540
6240 633
6249
708
79a
8768
9607
o446
1285
2124
2962
38oi
4639
5477
63i5
7i53
8828
9665
o5o2
1339
2176
3oi2
3849
4685
5521
6357
7193
8029
8864
9616 9699 9783
oaa3
1067
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2756
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13
2
2973
38i3
4653
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7173
8012
8852
o53o
1369
2208
3o46
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556i
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8074
891 1
9749
o586
1423
2259
3096
3932
4769
56o5
644i
7276
8112
8948
o3o7
Il52
1996
2841
3685
4529
5373
6217
7060
7904
8747
1276
2118
2961
38o3
4645
5487
6329
7170
8012
8853
1376
2217
3o57
38o7
4737
5577
64i7
7257
a
8
9775
0614
1453
22
3
1291
li3o
3
4606
5645
6482
7320
81 58
8oq5
2
0669
i5o6
2343
3i8o
4oi6
4852
5688
6524
736o
8106
90 J I
9866
3081
2925
3769
46i3
5457
63oi
7145
7988
883 1
9674
o5i7
t36o
220a
3o45
3887
4739
5571
64i3
7254
8096
8937
9778
0619
i46o
23oi
3f4i
3981
4821
566i
65oi
7341
8180
9020
985o
0698
1 537
2375
32i4
4o52
4890
5728
6566
74o4
8241
9079
1590
2427
3263
4i<)o
4936
5773
6608
7444
8379
911$
9950
01234 56789
10
Die
84
Diff.
Diff. & MttlL
a
3
4
5
6
7
8
1
a
3
4
5
6
7
8
I
a
3
4
5
6
7
8
85
170
a55
340
4a5
5io
595
680
765
84
168
a5a
336
430
5o4
588
673
756
83
166
349
333
4i5
498
58 r
664
747
DiCAtMoIt
74
LOOABITHMS OF NUMBERS.
Ninii*
ttlOO
01
03
o3
o4
o5
o6
07
08
09
53 10
II
13
i3
i4
i5
16
17
16
»9
53 30
31
33
33
M
35
36
37
36
39
53 So
3i
33
33
34
35
36
37
38
39
5340
4i
43
43
44
45
46
47
48
49
53 5o
5i
53
53
54
55
56
57
58
59
Nurau
12 3 4
5 6
TI6oo33
0860
i7o3
3538
3373
4307
5o4a
5876
6710
7544
8377
9311
7170044
0877
1710
3543
3376
4308
5o4i
5873
6705
7537
6788
7620
8369 ^453
9300
7180033
08630946
1694 1777
3535 3608
3356 3439
418643694353
5oi7
5847
6677
7507
8337
9167
7190026
i655
3484
33i3
4i43
4970
7455
8383
91 1 1
• 9938
7300766
1593
3430
3347
4074
4901
5737
7380
8306
9033
0117
0953
1787
3633
3456
4391
5i35
7637
846i
9394
0137
0961
1794
a6a6
3459
4392
5i24
5956 6039
9383
oii5
5ioo
5930
►079 oi6
0900009
1736 182
7538
9193
♦031
0848
1676
a5o3
333o
4i57
4983
58io
5893
655466366719
7545
8371
>i97
7462
8388
911
oaoo
io36
1870
37o5
3540
4374
5308
6o43i6i36
68776960
7794
7710
8544
9377 946
03II
1044
1877
0384
II 19
1954
3780
3633
4458
5393
8637
I
0394
1137
i960
2710
3542
4375
520715290
6133
2793
3626
4458
6871
77o3
8535
9367
0198
1039
i860
3691
3522
5i83
6oi3
6760 6843
75907673
8430 85o3
9350I9333
"2
3
1021
365o
173
2567
3396 3479
4324
5o53
588 1
6710
43o7
5i36
5964
6793 6875 6958
7631
83668448
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0931
1758
2586
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4239
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6955
7786
8618
9450
0281
1113
1943
3774
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5366
6096
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7756
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9416
0245
1075
1904
3733
3562
4390
5319
6047
77o3
853 1
9276 9359 944
0187
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2668
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4322
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5075
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7628
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6309
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7877
8711
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0377
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2043
2876
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6206
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1195
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6179
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1087
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3
0269
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1934
2751
3578
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6o58
6884
7710
8536
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1386
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6393
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9637
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1370
2304
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47084791
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6376(646o
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796080448137
87948877896
9711
1 377
2310|33i
3o43
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5540
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63i3
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1 179
3007
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4487
53i4
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6967
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7131
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3309 3293
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8835
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04940577
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3080
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8045 9038
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3337
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4818
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8949
9775
01234 56789
83
Die
DUL&lfniL
I
84
3
168
3
353
4
336
5
430
6
5o4
7
588
8
673
9
7i6
I
8,3
3
166
3
349
4
333
5
4i5
6
7
^
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L0GABITHM8 OF IfUMBEES.
75
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1109
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4297 4379
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2801
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1189
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2832
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6116
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6585
7402
8918
9035
01234 56789
Dim
IMC fc Malt.
t
2
3
4
5
6
7
8
82
Diff.
I
3
3
4
5
6
7
6
I
2
3
4
5
6
7
8
9
83
t66
249
332
4t5
664
747
89
i64
246
328
410
492
574
656
738
81
162
243
324
4o5
486
567
648
729
Dtff.lt Halt.
76
LOOARITHMS OP NXTICBE&S.
Nom.
5330
21
22
a3
a5
a6
a8
29
53 3o
3i
3a
33
34
35
36
37
38
39
5340
4i
4a
43
44
45
46
4?
48
49
53 5o
5i
5a
53
54
55
56
57
58
59
53 60
61
6a
63
64
65
66
67
68
69
5370
71
7a
73
74
75
76
77
78
79
Num.
1234 56789
72591169198
• 9933i«oi4
726 07490830
1 565 1646
a38o
3196
4oi2
4827
5642
6457
7372
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727 o53o
1 344
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2972
3786
4599468
54i3
6226
7039
7852
8664
• 9^77
728 0290
1 102
1914
3726
3538
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5i6i
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6784
7595
84o6
9216
729 0027
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730 o552
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2977
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701 5
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1728
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4173
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7354 7435
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63696450
7177I7258
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2707
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1427
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3863
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7108
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8730
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1972
3783
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7640
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9a58
♦066
0875
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2492
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4io8
4916
5723
653 1
7338
9524
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1973
2788
36o4
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523
60 5o
6865
7679776
8494
9300
01 2J
0937
175 1
2565
3379
4192
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2320
3i32
96069688
0422
1238
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2870
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53 16
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2647
3460
4274
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5901
6714
7527
8339
9152
9965
0777
1589
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32x3
39444025
4755
5567
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8000
8811
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4189
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1 324
ai34
2944
3754
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7028
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1914
2728
3542
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5982
6795
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1670
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7270 735
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2809
3623
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1752
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6621
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0675
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3396
3io6
3916
4726
5535
6345
7i54
7963
8772
0390
1198
2007
2815
3623
443 1
5239
6o46
5o77 5 1 58
58855966
6692677316854
75001758117661
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4745
556i
6376
7191
8oo5
8820
9634
0449
1363
2077
3891
3704
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6958
7770
8583
9396
0308
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3645
3457
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90549135
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0757
1567
2377
3187
3097
4807
56i6
64a6
7335
8o44
8853
9663
0471
1379
3088
2896
3704
45i3
5330
6137
6935
774a
1234 56789
DiC
81
Dur.
DULttMOL
1
a
3
4
5
6
7
8
9
8a
164
346
338
410
493
574
656
738
I
81
3
163
3
343
4
3a4
5
4o5
6
486
7
56?
8
648
9
739
Dur.
JtUviL I
LOaARITHMS OF NUMBERS.
77
Nank
8i
8a
83
84
85
86
87
88
»9
5390
$
97
98
99
5400
01
02
o3
04
o5
06
07
08
09
54 10
II
13
i3
i4
i5
16
17
18
»9
5420
ai
aa
a3
34
a5
a6
^
a9
54 3o
3i
3a
33
34
35
36
37
38
39
Nob.
12 3 4
730 7833,7903 7984 8o65 81^
863o
• 9437
73i oa44
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5i6a
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4276
5o8a
5888
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83o4 8385
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• 9914
73a 0719
i5a4
a3a9
3i33
3938
4742
5546
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7153
/
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8760
• 9564
733o367
1170
1973
2775
3578
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8390
919a
•.9993
1565
3396
3197
4798
5598
6398
7198
29^
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• 9598
73503970477
1196
1995
2794
3593
8791
8711
9518
o3a4 0465
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1938
2744
355o
43564437
596860496
67746854
7660
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9370
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7234
8o37
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3856
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5a63
0003
8470
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♦073
734 0794 0874 0954
2476
3277
4077
4878
5678
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7378
8078
8878
9678
1376
3075
3874
3673
439314472
5191 5370
596967997^
4o5 04860567
1313
2018
28a5
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5243
♦075
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2490
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0139
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5707
4179
4983
5787
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73147394
8118
8931
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2i33
2936
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4541
5343
6145
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01 53
1755
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5324i54o4
209?
2905
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8198
9001
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0608
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2313
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68676947
7669 7749I7839I7909
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1 835
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4238
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6718
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83x8
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55x0
5 6 7 8 9
8226 83o7 8388 8468 8549
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9033
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4679
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7096
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161 D X696
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39x7
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2715
35x3
43x2
5xxi
80
DiCfclfah.
IT"
x
a
3
4
5
6
7
8
9
I
a
3
4
5
6
7
8
9
I
a
3
4
5
6
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243
324
4o5
486
567
648
729
80
160
240
320
400
480
56o
64o
720
01234 56789
Dili:
M
237
3x6
395
474
553
63a
7^
DiAAdfoh.
78
LOOAIITEKS or IfUMBSKS
Nan.
6440
4i
4a
43
44
45
46
47
48
49
54 So
5i
5a
53
54
55
56
57
56
59
54 6o
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62
63
64
65
66
67
68
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73
74
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76
77
78
79
54 80
81
8a
83
84
85
66
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88
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5490
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9?
93
94
99
Unn.
12 3
T35 5989 606916 1 49
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8383
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1 574
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1733
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33a7
4ia4
4921
5718
65i4
73ii
8107
8903
0494
1390
ao86
a88i
3676
4471
5a66
6061
67766855
7570 765o
83648444
9159 9a38
♦o3a
o8a6
9953
0747
i54o|i6ao
241 3
3ao7
4ooo
475
558
6379
717a
7885 7964
89669045
9756 9835
0625
i4i5
2ao5
aooS
6a a8
7oa7
78a5
86a3
94ao
oai8
1016
idi3
a6io
3407
4204
5ooi
5797
6594 6674
7390
8186
898a
o574
1370
ai65
3960
3755
455o
5345
40794
4872
5665
6458
7a5i
8043
8836
0420
laia
8756
9549 96a8
o34i
ii33
i9a5
a7i6
35o8
4299
5091
588a
6673
7463
8a54
aoo4
2796
3587
4378
5170
5961
675a
7543
8333
9ia4
99«4
0704
1494
aa84
3o74
4
sssr
7107
7905
870a
9500
0298
1095
1893
3690
3487
4a84
5o8o
5877
5 6 7 8 9
Diff.
DUE ^Mah.
7470
8a66
906a
9858
o654
1449
aa45
3o4o
3835
463o
54a5
8346 8425
914a 9aa I
9937 #017
0733 081 3
i5a9 1608
a3a4a4o4
31193199
3914*
4709^4789
55o4""
4o|daao
7oi4
7808
86o3
9397
019T
0985
1778
257a
3365
159
495a
5745
6537
733o
8ra3
8915
9707
0499
1391
ao83
a875
713666
4458
5a49
6a4o
683 1
76aa
84ia
9ao3
0783
1573
a363
3i53
6388
7186
7984
878a
9580 9660 9740 9819
0378
1 175
197a
2770
3567
4363 4443
5 160 5a4o
5957603661
67536833
7549(7639)7709
7094
7886
868a
i858
a65i
3445
4a38
5o3i
58a4
6617
7409
8aoa
0578
1370
ai6a
2954
3745
4537
53a8
611
6910
7701
84oi
9a8a
♦07 a
086a
i65a
a44a
3a3a
646816548
7a66
8064
886a
0457
ia55
ao5a
a849
3646
994
789
5584
6a99 6378
7173
7967
876a
947695569635
oa7o|o35o
1064
ii43
1937
373 1
35a4
o658
i45o
aa4i
3o33
38a4
46i6
5407
96
6108
6989
7780
8570
9361
oi5i
0941
1731
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7346
8144
894a
0537
i335
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3?^
45a3 460a
691a
85o5
9301
0097
089a
1688
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4074
4868
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8047
884t
o4ao
laaJ
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4396
7568
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9153
9866I9945
0737
1 539
a3ai
3iia
3904
4695
5486
6377
7068
9440
oa3o
loao
1810
a6oo
3390
6638
7436
8aa4
90a a
06170697
i4i4
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3009
38o6
6^
690a
7788
85848664
93809460
0356
io5i
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0176
0972
1767
3563
3358
4i53
4948
5743
o5o8
i3o3
3006
2880
368
t
4476
5369
6062
67756854
933 a
♦oa4
0816
1608
3406
3191
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7859 7938
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7147
7938
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9519
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83o4
9101
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3391
3o88
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637
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4a33
5o37
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81 a6
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6617
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8ao6
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3175
3763
4555
5348
6i4i
6934
7726
7647
844o85i9
9311
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1687
3479
3370
4n63
4853
5644
63566435
7336
8017
8808
9598
o3o9o388
1009 1 178
1889 1968
3679 ^758
3469^3548
I
3
3
4
5
6
7
8
80
160
340
3ao
400
480
56o
64o
720
I
a
3
4
5
6
7
8
9
a37
3i6
395
474
553
63i
7"
79
01234 567891 Difli
DiC^Jfvh.
LOGARITHBCS OF NUMBERS
79
N«D. 12
S5O0
oi
02
o3
04
o5
06
07
06
09
55 10
IX
la
i3
i4
i5
16
17
18
'9
55 ao
ai
aa
a3
U
a5
a6
*7
a8
»9
55 3o
3i
3a
33
34
35
36
3?
36
39
5540
4i
4a
43
44
45
47
48
49
5550
5i
5a
53
54
55
56
57
58
59
Km.
740 3627
44164495
5ao6
67S
7573
836a
9i5i
741 0738
i5i6
a3o4
4668
5455
6a43
7o3o
7817
8604
0177 oa
646665446623
7a5i
8o37
8822
• 9607
7430392
1176
1961
3745
353o
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5098
8a32
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744o502
i365
a 147
a
3712
4495
5277
3706
5285
6074
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2462
3a5o
♦018
0807
I
a
3171
39594037
47464825
733o
8ii5
8900
9685
0470
1255
2o39
2824
36o8
4I4392
5176
5960
5254
6o38
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7605
8389
5882
7449 7527
83ii
90169094
7999^9955
56206600738
0660
,443
2226
93o3oo8
3o86
3791 3869
4573 465i
5355 5433
6059I61376215
684x691
76227701
84o48482
9185 9264 9343
9967l«o45oi23
3_
3785 3864
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5364 5443
5 6 7 8 9
61 53
694a
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7810
85c
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6176
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1 333
21x8
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7266
8o53
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9627
o4i3
1200
1986
2773
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5916
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27
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7075
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8638
9420
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5522
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7100
7880
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2619
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9250 9329
♦o34
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1678
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4729 4807
5590
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48x1
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6390
7179
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0784
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5?5?
6548
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80478125
4890
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7258
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1 201
1989
2777
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06490728
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8702
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425? 4338
5o4o5x27
5837 5916
6636 6705
74x5
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2935
3722
45xo
298
6o85
6872
7660
8447
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7094
7880
8665
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1019
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2588
3373
4x57
494x
5725
65o8
729a
8076
191
269
3478
426o|4338
5o42
5824
6606
9643
0425
1208
1991
2773
3556
4338
5X20
5902
6684
7466
8248
9029
98x0
0592
7494
8283
9072
9860
0649
1437
aaa5
3oi3
38oi
4!
5:
6x64
6951
7738
8525
9312
1672
2458
3a44
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48x6
56o2
7x73
7958
8743
9450 9528
o3i3
1098
1882
2667
345 1
4235
5019
58o3
6587
7370
81 54
88598937
9720
o5o3
1286
2069
2852
3634
44x6
6762
7544
8326
9107
9889
0670
12 3 4
5 6 7 8 9
Dlir. k. Malt
Difl:
I
a
3
4
5
6
7
8
9
I
a
3
4
5
6
7
8
9
a37
3x6
395
474
553
63a
711
78
i56
334
3ia
390
468
546
6a4
70a
Diff.^Miilt.
I
80
LOOASITHM8 OF NUMBERS.
5560
6i
61
53
64
65
66
67
68
69
5570
71
7a
73
74
75
76
77
78
79
55 80
81
83
83
84
85
86
87
88
89
5590
91
95
$
99
56 00
01
oa
o3
04
o5
06
07
08
09
56 10
II
13
i3
U
i5
16
17
18
19
01234 56789| wc
^45 0748
1 539
sSio
3091
3871
4653
543a
6313
6993
777a
8553
• 9333
746 01 1 1
0890
1670
449
i33d
333
4006 4o84
4785
5564
6343
7130
7898
8676
7198
7976
8754
• 94549533
747 oa33
1009
1787
3564
334i
4ii8
4895
5673
7^35
8001
8777
• 9553
7480339
1 105
1880
3656
343i
4ao6
4981
5756
653 1
73o6
8080
8854
o4oS
749 o4«
1177
1950
2724
3498
4271
5o44
58i7
6590
0836
1607
3388
0904
i685
3466
3169(3347
39494037
4730480848864964
55io
6390
7070
785o
86308708
9410
0189
0968
1748 1835
3537
33o6
4863
564i
643064986575
o3i
1087
1864
3643
34i
oo387
93497
4196
4973
574
64486531
73o3
I
807
885
963i
o4o7
1 183
1958
3733
3509
4384
5o59
5834
6608
7383
8157
8933
o48o
1354
3038
3801
3575
4348
5l33
5
5588
6368
7148
7938
9487 9565 9643
0367
io46
36o5
3383
4163
4941
571
91 5797
7376 7354
8o54
8833
9610
o465
1343
3030
27191^797
ii65
1943
3o35
a8ii
3586
436i
5i36
746o
8335
97069783
0557
i33i
3io5
3879
3653
4436
597a
0983
1763
3544
3335
4io54i83
5666
64466534
7336
8006
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o345
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1903
3683
3461
4340
3760
3539
43i6
50195097
8x33
9687
4273
5o5o
5837
66o3
738o
8i56
893a
9708
o484
1360 1337
3Il3
3888
3664
5ai4
5911 5989
6686 &763
676
7538
83i3
90099087
9861
o635
1409
3i83
3956
3730
45o3
51995376
1060
i84i
363 a
3463
5744
73o4
8084
0433
I303
I981
3574
435i
5i38
5905
668
7458
8334
9010
97869863
o563
5875
6653
743i
83IO
8987
9765
0543
1 330
3098
3875
3653
4429
5306
5983
675
753
83ii
9087
9863
0639
i4i5
3190
3966
3741
44394516
5391
6066
684i
7615
83oo
9164
9938
0713
i486
3360
3o34
38o7
458o
5353
6137
6049
68336899
ii38
1919
3700
3481
4361
5o4a5i2o
5833 5900
66036680
73837460
8163 8340
I3I7
1998
3778
3559
4340
8942
9721
o5oi
1280
3059
2838
36i7
4396
5i74
5953
6731
680968876965
75097587
8387
9065
9843
0631
1398
3175
2953
3730
45o7
5283
6060
6836
7613
8380
916$
9941
0717
1492
2368
3o43
06980776
1476 "
3353
3o3o
38o7
♦016
0700
i564
2337
3iii
3884
4658
543 1
6304
6977
0579
1 358
3i37
3916
3695
4474
5a5a
6o3i
1395
3076
3856
3637
44x8 4496 4574
5198
5978
^58
7538
9030
97999877
8365
9143
9931
4584
536i
4663
5439
6i38|63i5
n4
8467
9343
♦019
0794
1570
3346
3l31
38193896
■ 4671
5446
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0867
i64i
341 5
3i88
3963
4735
55o8
6381
7054
0657
1436
33l5
2994
3773
4552
533o
6108
1373
2i54
2934
3715
5276
6o56
6836
7616
83188396
909891769254
7665
8443
9231
9998
1 554
333 1
3i
3885
o83
5534
6331
6996
7770
8545
9319I9396
6390
7073
7848
8633
0170
0944
1718
2493
3266
4o39
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5585
6358
7i3i
9955
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3393
307a
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4630
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6186
1457
333a
3oi3
3793
5354
6i34
6014
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8531
929?
♦076
8544
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3198
3974
47494836
56oi
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3939
4707
5486
6364
7042
7831
8598
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01 54
0933
1709
3486
3363
4o4o
4817
5594
6371
7147
7923
74 035 1
1037
i8o3
3578
3354
4139
49o4
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63766453
7i5i 7338
8oo3
8777
9551
o335
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2647
3430
4194
4967
5740
65i3
7386
01234 56789
DiClfclCidL
1
a
3
4
5
6
7
8
9
Diff.
I
a
3
4
5
6
7
8
iS
237
3i6
395
474
553
63a
711
I
78
a
i56
3
334
4
3l3
5
6
ig
7
546
8
634
9
703
77
i54
33l
3o8
385
463
6^
IKfilklliilt.
LOGARITHMS OF N U M B £ & 8 .
61
.
5«30
31
aa
a3
a5
26
a?
29
56 3o
3x
3a
33
34
35
36
37
38
39
56 4o
4i
4a
43
44
45
46
47
48
49
56 5o
5i
5a
53
54
55
56
57
58
59
56 6o
6i
6a
63
64
65
66
67
68
69
5670
71
73
73
74
75
76
77
7«
79
Nua.
749 7363
8i36
8908
• 9681
7500453
iaa5
J997
2769
3541
43ia
5o84
5855
66a6
73o8
8168
8939
• 9710
75xo48o
xa5i
aoai
1
7440
3 4
9°i
9787
o557
1 328
3098
2868
3638
4408
5i77
58705947
3791
356i
433 1
5ioi
663967
ia3a
3533
429*J
5o65
583i
6506
736a
8136
8893
754o4a4
1189
1954
2719
8ai3
8986
9758
o53o
i3oa
3074
2846
36i8
4390
5i6i
5932
6704
7475
8a46
751817593
8aooj8368
900319140
9835:9913
0608
69093
5a38
6010
6781
755a
83a3
16
7486
8255
9oa4
9793
o56i
i33o
3098
2867
3635
44o3
5i7i
7474
834i
9008
9775
o542
i3o9
2076
2842
3609
4375
5x4i
5907
6673
7439
8204
9864
0634
i4o5
2175
2945
3715
4485
5254
6024
6793
7563
8332
9101
98709946
8970 9046
96599735
o5oo
ia65
2odo
3795
o638
1407
3175
3944
3713
4480
5348
6or5
6783
755o
83i8
9085
9853
0619
i386
3l53
291
368
4452
5218
5984
6750
75i5
8281
9812
o577
1 342
2107
2872
o685
1457
3339
3ooi
3773
4544
53i5
6087
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I
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DifTltMalt.
82
LOGARITHMS OF HUKBBBS*
1234 56789
5680
8i
8a
83
84
85
86
87
88
89
5690
9'
9a
98
99
5700
01
oa
o3
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06
07
08
09
5710
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la
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U
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16
17
18
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5720
21
aa
a3
24
a5
26
27
28
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5730
3i
3a
33
34
35
36
37
38
39
754 3483
4248
5oi3
5777
6541
73o5
8060
883a
• 9596
55 0359
755
iia3
1886
a649
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4175
4937
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6462
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7987
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8642
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092 a
168^
a44a
3201
3960
4719
5479
6237
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9272
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oa38
0998
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47?5
5554
63i3
7072
783 r
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9348
0106
0864
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3i38
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8221
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2100
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5547 5624
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93589434
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1643I1719
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9651 9727
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2683
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57 1 3| 5780 5864
64706540I6621
7227 73o3 7378
798480608135
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6086,616a
6845 6920
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83628438
912019196
9878I9954
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01234 56789
DilLJtMidL
Dir.
1
77
2
i54
3
23l
4
3n8
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,385
6
462
7
8
6i6
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693
I
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2
152
3
a 28
4
3o4
5
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6
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8
608
9
684
Diir.JfcMalt.
LOGARITHMS OF NUMBERS.
83
12 3 4
5 6 7 8 9
5T40
4i
4a
43
44
45
46
47
48
49
5750
5i
53
54
55
56
5?
58
59
5760
6t
62
63
64
65
66
67
68
69
5770
71
72
73
74
75
76
77
78
79
5780
81
81
83
84
85
86
87
88
89
5790
91
98
99
758 91 1
• 9875
7590632
1 388
3i44
2900
3656
44i2
5i68
5923
919
66786754
7509
7434
8189
8944
• 96999/74
7600453
1208
1963
2717
3471
4225
6486
5o54
58o8
6562
734o|73i5
8068
87468822
95009575
761 6253
ioo5
1758
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3263
4768
5520
6272
7024
7775
8527
9278
7620030
0781
1 532
2283
3o34 3
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4535
5285
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9035
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4016 4091
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98
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211J
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1
2
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6
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8
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1 52
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532
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375
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525
600
675
74
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222
296
370
444
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DifllJcMalt,
n
84
LOGARITHMS OF NUKBERS.
N
6800
oi
o3
o4
o5
06
07
08
09
58 10
II
13
i3
i4
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16
17
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ai
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23
24
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a6
27
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32
33
34
35
36
37
38
39
58 4o
41
4a
43
44
45
46
47
48
49
58 5o
5i
5a
53
54
55
56
57
58
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Nun.
12 3 4
5 6 7 8 9
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7274
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3
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313
4
384
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Diff.JblMt
88
LOGARITHMS OF NUMBBSS.
MaHi.
6040
4i
4a
43
44
45
46
47
48
49
6o 5o
5i
59
53
54
55
56
57
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63
63
64
65
66
67
68
69
6070
71
72
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74
75
76
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78
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99
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791
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LOGAEITHMS OP NUMBERS.
89
0100
01
oa
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07
08
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61 10
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la
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53
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185 3398
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90
LOGARITHMS OF MUMBSKS.
Nob.
6160
6i
62
63
64
65
66
67
68
69
6170
72
73
74
75
76
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I092lii58
57561 58aa
64i5648i
70747140
7734 7800
8393 8459 8525
9118
9777
03700436
0240
08980963
i555 1 020
2212
2869
3525
4182
54q5
61 32
6808
7464
8iai
871 1 8777
93679433
♦0230089
06790744
i334 i4oo
2o55
271 1
3366
4022
4677
1094
1753
2411
3070
3728
4386
5o45
5703
636o
7018
7676
8334
8091
3277
2934
3591
4248
43394905
556i
6218
6874
753o
8186
884a
9498
01 54
0810
i465
2121
2776
3432
4087
474a
5397
6o5a
6707
7362
8017
4^^
5228
5888
6547
7206
7866.7
397514041
4634 4700
52941 536o
59536019
66136679
7272
931
9»84
9843
o5oi
1 160
1819
^77
3i36
3794
4452
5iio
5768
6426
7084
7742
8399
9057
97>4
0372
3343
3ooo
3657
43i4
4970
5627
6283
0220
0875
i53i
3187
2842
3497
4i53
4808
5463
6118
6773
7427
8082
8737
♦045
0700
x354
2787
3447
4107
4766
5426
6o85
6745
7338 7404
7997I8063
8656 8722
92509315
"9974
2543
3 20a
386o
45i8
2609
3267
3926
4584
51765342
5834
649a
7i5o
7808
8465
58
7216
7873
6S3i
91239188I9254
97809846
0437 o5o3
1095 1 160
1732 1817
2409
3o66
3723
4379
5o36
5528
6i83
6838
7493
8147
8802
a474
3i3i
3788
4445
5ioa
5758
64i4
7071
7727
8383
II
1007
1662
9381
♦o4o
35i3
4173
4832
5492
6i5i
681 1
7470
8129
8788
9447
0J06
0765
1424
2016I2082
2741
3399
4o57
471 5
5374
6o32
6689
7347
8oo5
8662
9330
9977
o635
1292
1949
2606
3263
3920
4576
5233
9105
9761
o4i6
1072
1728
5594
6249
6904
7558
821 3
8867
5659
63i4
8278
8q33
9456 9522 9587
017610242
o83o
i485'i5do
2540
3197
3854
45ii
5 167
58a4
648o
7i36
7793
84498514
9170
9826
o48a
ii38
1793
2449
3io4
375?
44i5
5070
5735
638o
7034
7689
8344
12 3 4
5 6 7 8 9
Oifl
66
Diff.
Dim A; Malt
I
3
3
4
5
6
7
8
I
3
3
4
5
6
7
8
9
66
l33
t
33o
538
594
65
i3o
io5
260
335
/
Sao
585
DiflllKMolL
13
98
LOOAEITHICS OF NTJKBEKS.
Noau
6640
4i
4a
43
44
45
46
47
4B
49
66 5o
5i
5a
53
54
55
56
57
58
59
66 6o
6i
6a
63
64
65
66
67
68
69
6670
71
7a
73
74
75
76
77
78
79
6680
81
8a
83
84
85
86
87
88
89
6690
9'
98
99
N«m.
12 3 4
5 6 7 8 9
833i68r
a335
a
4396
4950
56o3
6a57
6910
7563
8316
8869
• 959 a
8a3oi75
o8a8
i48i
ai33
a786
3438
4090
474a
5394
6046
800a
8653
93o5
8a4
xa58
I
3aii
386a
45i3
5i63
58i4
6464
7114
7765
84i5
9065
• 9715
8a5o364
1014
1664
a3i3
a963
36ia
4a6i
4910
555o
6ao8
75o6
8i54
88o3
• 9451
8a6oioo
1746
a4oo
3o54
3708
436a
5oi5
5669
63a a
6075
7639
838 a
8935
9S88
oa4i
08930958
1 546
3198
a85i
35o3
4i55
4808
5460
6111
6763
741 5
8067
8718
93709435
0086
0737
783o
848o
9i3o
9780
1079
1730
2376
3oa8
3677
4336
4575
5634
6373
6857 6933
7571
8319
8868
9516
oi65
l8l3
3466
3119
3773
44a7
5o8i
5734
6387
7o4i
7694
8347
9000
9653
o3o6
161 1
3364
30l6
4331
916 396
568 363
4873
55a5
6177
68a8
7480
8i3a
8783
♦03 1
0673
1 333
1075
3035
3376
3937
4578
5338
5879
6539
7179 7344
1369
304o
3691
334i
3993
4643
5393
5o44
6594
7895
8545
9845
04390494
T144
1794
3443
3093
374a
4391
5o4o
5689
6336
6987
7636
8384
8933
9581
03391
5i46
7106
7759
84i3
9065
9718
0371
ioa4
1676
3339
34
4a86
4938
5590
634a
6894
7545
8197
8849
9500
oi5i
o8o3
1454
aio5
3756
34u6
4o57
4708
5358
6009
6659
73io
7960
86ro
9360
ro
0559 063
1309
i85o
35o8
3i57
3807
4456
5io5
5754
64o3
7053
1877
353 1
3i85
3839(3904
449a 4558
531 1
5865
65i8
7171
78a5
8478
9i3i
9784
o436
1089
174a
3394
3o47
4I??
5oo3
5655
63o7
6959
7611
8a6a
8qi4
9565
03 16
0868
i5i9
3170
3831
347a
4iaa
4773
5433
6074
67a4
7375
8oa5
8675
9335
a4
1374
1934
3573
3333
387a
45ax
5170
5819
6468
7117
7765
84i4
906a
971 1
0359
aoo8
3663
33i6
5377
6583
7337
7890
8543
9196
9849
o5o3
ii54
1807
3459
3lI3
3764
4416
5o68
5730
6373
7034
7676
83a7
19
0383
0933
x584
3335
a886
3537
4187
4838
5489
6139
8740
9390
♦o4o
5a35
5884
6533
7181
783o
9137
9776
3073 3i3oj2ao4,3369
37931
3737
338i
4o35
4668
5343
5
496714
73o2
7955
8608
9361
991 4
0567
1330
1873
3535
3177
38393894
448i
5i34
5786
6437
7089
774 1
839a
4547
65o3
7i54
7806
8458
9<V<4 9109 9174
9695
o347
0996
1649
33oo
^'
36o3
4a53
490^
555^
3
54
6304
6789 6854
744o 75o5
80908155 8aao
I
88o5
93909455
oio5
06890754
i4o4
3o53
3703
3353
40O3
1339
Z988
3638
3387
3937
4586465
53oo
6
7346
7895
84798544
9193
9840
04340489
3858 3033
3446;35i3 3577
4 1 00: 4 166 4a3 1
47544819
5407
6061
7367
8oai
8674
9327
9079*045
oo33 0697
1385
1937
3390
334a
9761
o4ia
io63
i7>4
3365
3oi6(3o8i
3667
43i8
6919
7570
i46o
aii8
3768
3417
4066
5365
6014
6663
73ii
7960
8608
9357
2?54
5473
61 36
6770
7433
8086
87398804
9392
i35o
3oo3
3655
33o7
3o6o
4013
4884
5538
6191
6845
7498
8i5i
9457
Olio
0763
i4i5
3068
3730
3373
4oa5
4677
5a64 5399
5916
6$68
7330
7871
8533
9836
0477
1138
5981
6633
7385
7936
8588
9339
9891
o543
1193
1779
343o
3733
4383
4968 5o33 5098
5619 5684 5749
636963346399
6084
7635
8a85 835o
8870 8o35 9000
95ao 9585 9650
0169 0334 0399
081908840949
1 534
3183
3833
3483
4i3i
471647804845
543o
6078
OOIi
8
i844
a495
3i46
3797
4448
7049
7700
1590
3348
3898
3547
4196
5494
6143
6737 6793
73767441
8o35
8673
9333
o
8090
8738
9387
♦o35
o683
01234 56789
Dim
DiCltMnlt.
I
3
3
4
5
6
7
8
9
65
66
1 33
33o
5a8
594
I
65
a
i3o
3
4
t
5
335
6
7
s;
6
530
9
585
Dilt Diir.JtUalt
LOGARITHMS OF NUMBERS.
99
12 3 4
5 6 7 8 9
0700
oi
oa
o3
o4
o5
o6
07
08
09
6710
II
la
i3
i4
i5
16
17
18
»9
6780
ai
aa
a3
U
a5
a6
rj
a8
29
6730
3i
3a
33
34
35
36
38
39
6740
4i
4a
43
M
45
46
47
48
49
6750
5i
5a
53
54
55
56
57
58
59
nBBk
07480813
1396! i46i
ao44 axo9
a69a
3340
2757
34o5
• 98ao
8a9o463
1107
1751
3394
3o38
368i
43a4
4967
56ii
6a54
0878
i5a6
ai74
a8aa
3470
4117
4700I4765
54i3
6060
6707
3757
44o4
5o5o
5696
634a
7o5a
7698
82798343
734
7887
853a
9i769a4o|93o4
8
9884
o5a8
1171
i8i5
9?4
o5o
I a
187
14591 a5a
3ioa
3745
4389
5o3a
5675
8a46
7355
8ooa
8640
9396
9943
ia36
i883
aSao
3176
38aa
4468
5ii4
5760
64066471
3440 35o5
4o85
473o
5375
601
9608
6738
737a
8016
8660
4i8a
483o
5477
6ia5
677a
7419
8067
8714 8778
9361
♦007
o654
i3oi
i?47
a594
3a4o
3887
4533
5i79
58a5
7117
776a
84o8
9053
oaSo
1570 1635
aai5
a86o
aaSo
3935
3i66
38io
4453
5096
5739
638a
7oa5
7668
83io
4a i4
4859
55o3
6i48
670a
7437
8081
8735
9369
♦01 3
o656
i3oo
1944
3587
333i
3874
45i7
5i6o
58o3
6446
7089
773a
8375
88a4 888918953 9017 9081
1007
i655
a3o3
9951
3$99
4a47
4895
554a
6190
6837
7484
8i3i
9435
0073
0719
i366
aoi3
a659
33o5
5a44
7181
7837
8473
9118
96999763
o344o4o9
■ io54
1699
a344
3989
35693634
568
4a7
4
5
6313
6857
75oi
8145
8780
9433
0077
0731
i365
aoo8
3653
3395
Sofe
4583
5335
5868
65ii
7i54
7796
84^9
107a
1730
3368
3oi6
3664
43ia
4
6354
690a
75401
8196
8843
9400
0137
0784
x43o
3077
37a3
3370
ii37
1785
2433
3o8i
3729
4376
5o34
5673
63 19
6966
7614
8361
9555
oaoi
o848
1495
3l4l
3788
3434
0913007!
1360 I03.
4oi
4663
53o8
5o54
6600
7346
7801
8537
9t83
9838
0473
1119
1764
2409
3o54
3698
4343
4988
5633
6377
6931
7565
8310
8854
9498
oi4i
0785
1429
3073
2716
335o
4ooJ
4646
5389
593a
64080
7318
7861
85o3
9146
4727
5373
6019
6665
73 10
7?56
860a
9247
9893
o538
ii83
1838
a473
3ii8
3763
4408
5o5a
5697
634i
.'3
3306
3853
3499
4i45
4791
5437
6o83
6739
7375
80a I
8a74
8
oao6
o85o
i4o3
31^7
3780
34a4
4067
4710
5353
5996
6575 6639 6704
7283
70a5
9310
730311267
i85olrQi5
a498'a563
3i46'3aio
3794
4441
5089
5736
6384
7o3i
7^8
8335
. »9
0366
8666
93i3
9957
0602
1248
I
25
3i83
3837
4472
5ii7
5761
6406
7o5o
8338
9636
0370
0914
1 558
aaoi
3845
3488
4i3i
4775
5418
6061
7346
7
9274
3858
45o6
5i54
58oi
6448
7096
7743
8
7
9684
o33i
1624
2271
2917
42104274
4856
55o2
6i48
67946858
743o
8o85
8731
9376944
♦022
0667
l3l2
1067
2003
3a47
3893
4537
5i8i
5836
6470
7114
775o
84o3
9047
9691
o335
X8
3
2266
2909
3552
548a
6125
6768
74"
8o53
8696
9338
i33i
»979
2627
3275
3923
4571
5a 18
5866
65i3
7160
7808
8455
910a
&
1043
1689
3335
3082
28
4930
5567
6212
75o4
8i5o
8795
I
6086
0731
1 377
2022
2667
33i2
4<
5246
7179
7823
8467
9111
9755
0390
1043
1686
233o
2973
36i7
4260
4903
5546
6189
6832
7475
8118
8760
^3
01234 56789
Diir.
DURIbMalt.
Dur.
a
3
4
5
6
7
8
9
65
i3o
'^
325
5ao
585
I
64
2
138
3
356
4
5
320
6
384
7
448
8
5l2
9
576
Diir.lt Molt
100
LOGAKITHMS OP NUMBERS.
Nam.
«T«0
6i
6a
63
64
65
66
67
68
69
6770
7'
7a
73
74
75
76
77
78
79
6780
81
8a
83
84
85
86
87
88
89
6790
9»
9a
93
94
99
6800
01
oa
o3
04
o5
06
07
08
09
6810
II
la
i3
i4
i5
x6
17
x8
"9
Nim.
12 3 4
5 6 7 8 9
• 9467
83ooro9
0762
i3o4
aoj6
a678
33ao
3962
4604
5a45
5887
6538
7169
7811
8452
9003
• 97^4
83 1 0375
1016
i656
2ao7
2037
3578
4ai8
4858
If??
83a
677
7418
8o58
8698
9337
0616
xa55
1895
a534
3173
38i2
445o
5o8o
5728
6366
70o5
7643
8a8i
• 955?
8330195
o833
1471
aioo
2746
3384
4oai
953i|95q5
oi74loa58
0816
i458
2100
274a
3384
4oa6|4<
4668
5309
5o5i
6592
7^54
7875
85i6
5i53
5792
643o
7069
7707
8345
8983
9621
02 r
1 535
2173
2810
3448
4o85
473^
536o
6634
0880
i52a
ai64
9aai
9862
o5o3
1144
1784
a4a5
3o66
3706
4346
9157
9798
0439
1080
1730
236i
3ooi
3642
4a8a
49aa
5563
6203
6842
748a
8iaa
876a
9401
♦o4i
o68o|o744
i3i9i383
4986] 5o5o
56a7
6267
7046
8186
88a6
9465
oio5
aoaa
a66a
33oo
5ai7
5855
64q4
7i3a
7771
8409
9047
9685
590323
08970961
2874
35ii
4149
4786
5423
6060
6697
727117334
9660
o3o2
0944
1587
aaa9
a87i
35i2
4i54
4796
5438
60796143
6721
736a
8oo3
8644
9285
^a6
o*
iao8
1849
2489
3i3o
3770
44io
63
o
10
8a5o
8890
9539
01
1447
ao86
3364
4oo3
464a
5a8i
5914
655<
7196
7835
8473
911 T
9749
o387
ioa5
i66a
a3oo
2938
3575
4^12
485o
5487
61 a4
6761
7398
9724
o366
1009
i65i
3293
2035
3d77
4218
4860
55oa
6785
7426
8067
8708
9349
I
1373
1913
3553
3iq4
3834
4474
5ii4
5755
6395
7034
7674
83x4
8954
95?3
oa33
087a
i5ii
ai5o
3789
34a8
4067
4706
5345
66a a
7a6o
7898
8537
9175
9813
o45i
1088
1726
3364
3ooi
363o
4276
4913
555 1
6188
68a5
746a
364T
4a83
4924
5
6372
6913
7554
8195
8837
9478
01 19
0759
i4oo
2o4i
a68i
33aa
396a
46oa
5a4a
9|5883
65a3
7162
7802
8442
22l4
2853
3492
4i3
477048
9852
o4o5
1107
»779l
2421
419?
mi
54o8
6047
6686
7334
7962
8600
9338
9877
o5i4
Il53
1790
2428
3o65
3703
43404404
5472
6111
6749
7388
8036
8664
93o3
9940
0578
X2l6
i854
2491
3129
3766
6336
6977
7618
8260
8901
9542
oi83
0823
i464
aio5
9145
9785
0424
1064
639|i7o3
2342
2081
3020
5536
6175
68i3
7452
8090
8728
9366
♦004
0642
1280
1918
2555
3193
383o
4467
5io5
5742
6370
7016
7653
0023
1265
iao8
255o
3192
3833
4475
5ii7
5758
64oo6464
7041
7683
8324
8965
96069670
0247
0887
1 528
2169
2809
345o
4000
47^0
5371
6011
665i
7 a
7
8570
56oo
6877
75i5
81 54
8702
94^0
0068
0706
i344
1981
2019
3256
38o4
453i
5x68
58o6
6443
7080
7716
♦045
0687
i33o
107a
2614
3256
3898
4539
5i8i
58a3
7io5
7747
8388
9029
2873
35i4
4i54
4794
5435
6075
6715
7354
71 _
4
200 7354
o3o7994
5708034
9209 9273
9849
0488
1128
1767
a4o6
3o45
3684
4323
4961
i3
5a
119a
i83i
2470
3109
3746
4387
5o25
5664
6239|63o2
" 6041
7379
8217
8856
9494
Ol32
0770
1407
2045
2683
3320
523a
5869
65o6
7143
7780
01234 56789
JHg.
64
Diir.
Diiribliidl.
I
a
3
4
5
6
7
8
I
2
3
4
5
6
7
8
9
65
i3o
325
520
585
64
128
192
256
320
384
448
5l2
576
Diir.«tMaiL
LOGARITHMS OF MUMBSES.
101
68S0
31
22
33
34
35
36
37
38
39
68 3o
3i
33
33
34
35
36
37
38
39
68 40
4i
42
43
44
45
46
47
48
49
68 5o
5i
53
53
54
55
56
57
58
59
6660
61
63
63
64
65
66
67
68
69
6870
71
72
73
74
75
76
77
76
79
12
7844
8480
9117
• 9754
834 o39o|o454 o5i7
1090 ii54
1736 1790
3363 3436
3o63
7449 7^1 3
80848148
8656187198783
9391
• 9926
835o56i
1196
i83i
3465
3ioo
3735
436o
5oo3
5638
6373
6906
7540
8174
8807
♦ 9441
8360075
0708
i34i
1975
a6o8
334i
3874
45o7
5i4o
5773
64o5
7o38
7670
83o3
8935
• 9567
8370199
o833
' 1463
3095
3737
3359
>33
5353
79077971
85448608
9181
9817
7OQQM02
36353698
4371
4
5543
6178
68i3
9064970
543 56o6
9354 9418
♦o53
9990
0635
1359
1894
3539
3i63
3798
4433
5067
5701
6335
8337
8871
95o4
oi38
0771
i4o5
3o38
3671
33o4
3937
4570
53o3
5836
6469
7101
7734
8366
8998
963i
0363
0895
1 537
3i58
3433
4o53
4685
53i6
8673
934419308
9881
4334
4<370
6341
^771^940
0688
1 333
1058
3593
3337
386i
44q6
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S764
7033
030I
o835
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3101
3735
4001
4634
5367
6533
7164
7797
9063
o336
0958
1390
3333
3790 3853
3485
4117
4748
5379I544
3
5555
5 6 7 8 9
1317
i854
3490
3i36
3763
43c8
5o34
7576
831 1
8846
9481
OII7
0751
i386
3031
3656
3390
3935
4559
5194
5838
63986463
7006
73o
766677
83oo8364
0365
3i65
3798
3368 3431
4607
533o
58995963
6595
7338
7860
84398493
969497579830
03890453
I03I
i653
3385
3917
4i8o
48ix
3
9373
♦008
0645
138l
1917
3553
3190
383^
446f
5007
5733
6368
7004
763o
8375
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045
0180
o8i5
i45o
3o85
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3988
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5891
6535
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9061
9694
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3338
3861
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40644137
4760
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6036
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7933
8556
91359188
io84
1716
3348
9173980
548 361 1
II
4^43
4874
55o6
8163 8336
8863
0073 oi36
07080773
1 345
108]
3617
3353
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5796
64^3
7067
77o3
8338
9609
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4o53
4686 4750
5330 5384
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4834
5456
6089
6733
7354
7987
8619
9351
9883
o5i6
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1779
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3o43
3674
43o6
4
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ao45
j68i
3317
4589465
5334
586o
64o6
7i3i
77667830
8465
0043
1 577
3313
3846
3481
4ii5
7333 7386 73
7930
8554
9188
9831
0455
1088
1731
3355
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3631
4254
4887
55 30
6i53
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8o5o
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o57^
I3II
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3io6
3738
4369)443
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3
5388
24
6550
7195
9100
9736
0371
ioo5
1640
3375
3910
3544
794343
134876
4i
48i
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6081
671
43o6
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5574
6308
6167796842
7340
7983
8617
9351
9885
o5i8
ii5i
1785
3418
3o5i
3684
4317
4950
5583
6316
6785 6848
7481
8ii3
8745
93149378
♦010
7906430705
38oi
3
5o64
5695
8353
9036
0363
4716
535 1
5587
6633
7358
7893
8539
9164
97'
8417
9053
9690
0337
8389
8936
9563
oioojosoj 0J37
o8j6 0899 0963
1473 1 536 i!
3108 3173 33,
3744 a8o8 3873
338o 3444 35o8
4oi64o8o4i43
1069
1704
3338
2973
36o8
4243
55ii
6145
74i3
8047
8681
9314
9948
o58i
T3l5
i848
3481
3ii5
3748
438 1
5oi3
5646
6911
7544
8176
8809
9441
0073
0705
1337
1374
1906
3538
3169 3333
3864
4495
5i37
5758
4779
54i5
6o5i
6686
7333
7957
8593
9337
9863
0498
Il33
1767
3403
3o37
3671
7476
8110
8744
9378
♦on
o645
1378
1911
3545
3178
38ii
4444
5077
5709
6379 6343
6075
7607
8340
8873
95o4
oi36
0768
i4oo
3033
3664
3396
3937
4559
5190
5831
ir«i. 01234 56789
Diff.
ItMoh.
Dir.
I
3
3
4
5
6
7
8
9
I
3
3
4
5
6
2
64
138
193
356
330
384
448
5l3
576
63
136
353
3i5
378
441
5o4
567
DUTIfcllalt.
102
LOOARITHMS OF NTTMBBSS
Nun.
•880
8i
8a
83
84
85
86
»7
88
89
6890
9»
t
99
6900
01
03
o3
04
o5
06
07
08
09
6910
II
la
i3
i4
i5
16
17
18
19
6920
ai
aa
23
a4
a5
a6
27
38
29
6930
3i
3a
33
34
35
36
37
38
39
Nam.
12 3 4
83T5884
65i6
7147
7778
8409 847a
90399103
9733
o364
• 9670
838o3oi
0931
1 56a
219a
a8aa
3453
4o83
47i3
5343
5973
660a
7a3a
7861
8491
91 ao
• 9750
8390379
xoq8
i637
aa66
2895
35a3
4i5a
4780
5409
6037
0000
7922
855o
9178
• 9806
8400433
1061
1688
23i6
2043
3571
4198
48a5
5048
6579
7a 10
7841
6011
664a
7373
7904
8535
9166
97969859
0437 0490
i6a5
aa55
3886
35i6
4i46
4776
5406
6o36
6665
7295
7934
8554
9183
981a
o44a
1 071
1700
3339
3957
3586
4ai5
4843
547a
io57
1688
33i8
2049
3579
73947357
7985
861 3
9341
9868
0496
iia4
1751
2379
3oo6
3633
4360
4888
5453|55i5
[4i
6074
6705
7336
7967
8598
9339
1121
1751
338i
3oi3
3642
4309 4372
4839 490a
553a
6161
6791
74a I
8o5o
5469
6098
6738
7358
7987
8617
9346
9875
o5o5
ii34
1763
3393
3030
3649
4278
4906
5535
6ioo6i63
6738 6791
" 741
1186
1814
3441
3069
3696
4333
4950
5577
6304
683 1
7458
8084
8711
9337
9964
0590
I3l6
1842
34063468
3o3i 3094
6137
6768
8o$o
8661
939a
22
od53
1184
1814
3444
3075
3705
4335
4965
5595
6aa4
6854
7484
81 13
8743
937a
♦ooi
o63o
1359
1888
35i7
3i46
3775
44o3
5o32
566o
63366289
6854^
8680
'0^
o
1197
1836
3454
3o83
3713
4341
9748a
804781108173
8675 8738
93o3
3 1
33
ia49
1877
35o4
3i33
3759! 38
4386
5oi3
564o
6367
7520
8147
8773
9400
♦026
0653
1279
1905
353
3i57
6917
7545
8801
93669439
♦057
0684
l3t2
4449
5076
5703
633o
68946956
7583
8310
8836
9463
0089
0715
i34i
1967
3593
3319
5 6 7 8 9
62oo|6263|6326 638916452
683 1
7462
8093
87248787
9355
;i6
1247
1877
25o7
3i38
3768
4398446
5028
5658
6387
6917
7547
8176
8806
9435
0064
0693
l322
1951
258o
a
320I
383 1
I
5091
5731
635o
6080
7010
8339
4466
5095
5733
635i
8336
8864
9493
on
0747
1375
aooa
a63o
3357
3884
45fi
5i38
5765
6393
7019
i6
373
8a7
8899896
q535 o58
i4o4
ao3o
3656
3a8a
6894
7535
8i56
9418
♦049
0679 ^42
i3io
1940
5702633
94'
0137
0756
i385
30l4
2643
3373
5i58
5786
64i4
7o4a
7671
8299836
8937
9554
82
0810
901
1437
ao65
2692
3330
3947
4574
530I
58a8
6455
708a
7708
8335
I
95a 5 9588
oi5aoai4
07780840
6o57
7588
8219
88508913
9481
0112
1373
2oo3 2o66
3264
3894
45a4
5i54
5784
641
7043
0I7673
83o2
I
9D01
0x90
0819
1446
2077
2706
3335
39003963
45294592
3|6476
7106
7736
8365
5220
5849
6477
7io5
7733
8089
9617
0245
0873
tSoo
3138
3755
3383
1467
3093
3719
3344
7771
8398
9034
9650
0277
7020
7652
8283
95449607
"oa38
0868
"499
2129
0175
o8o5
i436
2696
3957
4587
5217
5847
2759
3337 3390
403O
465o
5280
5910
6539
779*'
8428
9057
9687
o3i6
0945
1574
8094
96a4
0353
088a
i5ii
2l4o|2203
283a
3460
1089
i7i8
3769
3398
4036
4655
9o5a
o3o8
0935
1 563
3190
2818
3445
4oxo 407a
4637
5264
5891
65i8
7144
5953
58o
7834
8460
I330
3i55
3781
3407
09030966
1330 1 503
1093
3318
2844
3470
7084
7715
8346
8976
Dur.
471
5346
5574
66o3
7a3i
5283
5912
654o
7168
77967859
84a48487
9115
96809743
0371
0998
1626
2253
a88x
35o8
4x35
476a
6016
6643
7207 7370
7896
8533
^49
>770
9087 914
97i3
o339|o4o2
1 038
i654
2280
3906
3532
DiClbMtfL
63
I
64
2
laB
3
4
X
5
330
6
384
7
448
8
5l2
9
576
I
63
a
ia6
3
189
4
252
5
3x5
6
376
7
44i
8
5o4
9
567
01234 56789
Dilt I Diltfclfott.
LOGARITHMS OF NIXMBBRS.
103
it
4a
43
44
45
46
47
46
49
6950
5i
52
53
54
55
56
57
58
59
6960
61
6a
63
64
65
66
67
68
69
6970
71
7a
73
74
75
76
77
78
79
6980
81
8a
83
S4
85
86
87
88
«9
6990
9»
99
12 3
8413595
4aao
48464909
547a
6097
67336785
7348
7973
8598
9aa3
• 9B48
84a 0473
1098
1732
a347
ao7i
3596
4a20
4844
5468
609a
671'
6677
7340 74
8
9311
• 9835
843o458
1081
1705
544a
6o65
73io
793a
8554
9176
9796
844o4aoo483
io4a
1664
aa86
aw
3539
4i5o
477a
5393
60U
6635
7a56
78777
84988
91199181
• 9739 9801
845o36oo4a2
3657
4a83
5534
6160 6aaa
7410
8o36
8661
938693489411
od35
1 160
1785
3409347a
3o34
3658
4383
4907
5531
6i55
ioS
8036
865o
3096
3731
4345
559?
6317
684i
7465
808918
9374 9336 9398
9897
o53o
II44
1767
3338
3951
3574
4i97 4a59|43
2
3oi3
3636
a
48194883
55o4
6137
6687674968
7373
7994
8616
9861
o483
iio4
9176 9339 93©
3348
3970
3591
4ai3
4834
5455
6076
6697
73i8
37ao
4346
4971
5597
6848
7473
8098
8723
973
0598
1323
1 847
8712
o
1206
1829
39o|3453
3075
3698
"31
4944
5567
6189
7434
8o56
8679874
I
9923
o545
1167
I7a6 1788
3410
3o33
3653
4275
489614
5517'
6i38
7380
8001
8622
9863
3782
4408
5o34|5o96
5659
6a85
6910
7535
8i6t
8786
♦o36
0660I0723
1 348
1072
2597
1285
1910
3534
3i59
3783
4407
5o3i
5656
6280
7^37
i5i
8775
♦022
0645
1268
1892
25i5
3i38
3761
4383
5oo6
5629569
6261
6874
7496
811
9363
9085
0607
1239
i85i
3472
3094
3715
4337
(958
5579
6201
6759682 a
7443
8o63
8684
4
3845"
4471
5 6 7 8 9
5733
6347
6073
7598
8333
8848
9473
0098
3331
3845
4470
5094
5718
6343
7:
8213
8837
9461
0084
0707
i33i
1954
2577
3200
3823
4446
5069
I
63i4
6936
7559
9818
9243 93o5
0026
88o3
9425
♦047
0669
1 291
1913
2534
3i56
3778
4399
5030
5643
6263
6884
75o5
8126
8746
9367
o484'o546
3
4533
4
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5847
64io 6473
51595
5784"
7o35 7098 7160 7323
76607723
8286
8911
9536
0160
07850848
i4io
3o35
3659
3384
4532
5i56
5780
64046467
7028
7652
8276
01460209
0770
1393
20 16
2639
3262
3885
45o84
5i3i
5753
7621
8243
8865
9487
0109
0731
i353
1075
2597
3218
384o
446i
5082
5704
6325
6946
7567
8f88
8808
9439
♦o5o
0670
of4o33
4658
5384
8348
8973
698
0333
90369098
147a
2097
272a
3346
^.o
5219
5843
7091
7714
8338
9661
0285
0910
1 535
2160
a784
3408
4o33
4657
5a8i
5905
65a
7i5
7777
84oi
9024
88998962
95a3 9585 9648
0271
08320894
1455 i5i8
20792141
2702
33a5
3o48
570
5193
58i6
6376 6438
7061
7683
83o5
8928
9550
0172
0794
i4r5
2o37
2659
3280
3902
4523
5i45
5766
6387
7008
7629769
8250
8870
9491
ori2
0733
653
7785
84ii
9723
o348
0973
1597
2222
3846
3471
4095
4719
5344
5968
0659
3721
a764
3387
4010
4633
5355
5878
65oo
7123
7745
8368
8990
9013
0234
o856
1478
2099
2721
3343
3964
4585
5207
5838
6449
7070
83i2
8933
9553
0174
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56
I
2
3
4
5
6
7
8
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57
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171
2a8
365
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01234 56789
Dur.
DiiriEltaU.
HBO
8i
83
83
84
85
86
87
88
89
7790
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93
94
95
96
98
99
7800
01
03
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04
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06
07
08
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11
13
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14
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16
17
18
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7830
31
33
33
24
35
36
37
38
39
78 3o
3i
33
33
34
35
56
37
38
39
LOGABITH
i 2 3 4
M8 OF NUKBERS.
5 6 7 8 9^
0913
1470
ao38
59431
49394984
5375|543o|S486 554a
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6489,6
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7604
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7660
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9331
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3616
3173
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• 9846
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3178
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1496 1 554
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8178
87348790
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1 558
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3338
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4807
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6010
6566
7133
7W
8334
9345
9901
0457
IOI3
i568
3133
3678
3334
3789
4344
7673
8338
8783
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0466
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1 583
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3090
3a56
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6044
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7i58
77x5
9387944
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1670
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6177
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1166
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01 1 1
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4007
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5l30
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1790
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7340 7396
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0689
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855i
9108
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4o63
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5176
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801 1
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1901
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3
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5343
5898
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7007
7563
8117
8671
9336
9780
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1443
355 1
3io5
01234 56789
Diff.
Difl*.
117
DUC 4b Mnlu
I
3
3
4
5
6
7
8
9
56
113
168
334
380
336
393
448
5o4
I
3
3
4
5
6
7
8
55
no
i65
330
375
33o
385
440
495
our. Ai Molt
118
LOGARITHMS OF NUMBERS.
Nam.
7840
4i
42
.43
44
45
46
47
48
49
78 5o
5i
5a
53
54
55
56
57
58
59
7860
61
6a
63
64
65
66
67
68
69
7870
71
72
73
74
75
76
77
78
79
7880
81
8a
83
84
85
86
87
88
89
7890
91
93
94
96
98
99
NOBI.
12 3 4
894 3i6i
3715
4a68
48aa
5376
592
648
7037
7590
8i43
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9130
• 9803
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9195
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i4o3
1954
a5o6
3o57
36o8
4160
471 1
5a6a
58i3
6364
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8017
8568
91 18
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5171
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220
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275
6
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7
385
8
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495
Dur.atiiBic
L09ARITH1C8 OF NUMBERS.
119
TMMI
01
o3
04
o5
o6
07
o8
7910
II
la
i3
i4
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16
17
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38
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7930
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43
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Dur.
DUr. k, Mah.
X
3
3
4
5
6
7
8
X
3
3
4
5
6
7
8
9
55
xio
i65
320
375
33o
385
440
495
54
108
163
3l6
270
324
378
433
486
Di&A&MaR.
/^
120
I.OOARITHMS OP NUMBERS.
NmB.
7960
6i
62
63
64
65
66
©7
68
69
7970
71
7a
73
74
75
76
77
78
79
7980
81
83
83
84
85
86
»7
88
89
7990
9'
92
98
99
8000
01
oa
o3
04
o5
06
07
08
09
8010
II
12
i3
i4
i5
16
17
18
»9
Nan.
12 3 4
9676973
900 91 3 1
901 0222
0767
i3i3
i858
a4o3
2948
3493
4o38
4583
5128
5673
6218
6762
7307
785 r
8396
8940
9485
9020020
0573
1117
1661
2205
2749
8293
3837
438 1
4924 4979
5468
6011
6555
7098
7641
8i85
8728
9271
• 9814
9o3o357
0900
1443
1085
2528
3071
36i3
4i56
4698
524i
5783
6325
686769
7409
7951
8493
9035
• 9^77
9040119
0661
1202
9185
I
6276
0822
1367
191 2
2458
3oo3
3548
40934
4638
5x83
5727
627a
6817
736 1
7906
845o
&
9
oo83
0628
1172
1716
2260
2804
3347
3891
4435
5522
6066
7i52
7696
8239
8782
9325
9868
o4ii
0954
1497
ao4o
a582
3i25
3668
4210
4753
5295
5837
22
7464
8006
8548
9631
0173
0715
1256
9340
9r785
o33i
08760931
1432
1967
2512
3o57
36o2
147
4692
5237
5782
6327
6871
74i6
85o5
9049
9594
oi38
0682
ia26
1770
23i4
2858
34o2
3946
4480
5o33
5577
6120
66096663
7207
775o
8293
8836
9923
o466
1008
i55i
2094
2637
3179
3722
4a64
4807
5349
5891
637964346488
6976
75i8
8060
8602
9144
9685
0227
0769
i3io
9294 9349
9840
o385
1476
202 X
2567
3lI2
3657
4202
4747
5292
5836
638i
69266980
7470
8oi5
8559
9104
9648
0736
1280
1824
2368
2912
3456
4000 4o54
4544
5087
563 1
6174
6718
7261
7804
8348
8891
9380 94 J4
9977
0520
io63
1606
2x48
2691
3234
3776
43x9
4861
54o3
5946
7o3o
7572
8xx4
8656
9198
974o
028 X
0823
i365
9894
o44o
0985
i53i
2076
2621
3x66
371 1
4256
4801
5346
75a5
8069
86x4
91 58
9702
0247
0701
1 335
1879
2423
2967
35ix
4598
5i4a
5685
6229
6772
73x5
7859
8402
8945
9488
♦o3x
o574
1X17
1660
2203
2745
3388
383o
4373
4915
5458
6000
654a
7084
7626
8x68
8710
9252
0877
1419
94o3
9949
0494 0549 ^^4
io4o
1 585
ax3o
3676
3aai
3766
43x1
485649
5401
5945
X04965
5455 5509
6000 6o54
65446599
70897144
7579 7634 7688
8124
8668
92x2
9757
o3ox
0845
'389
1933
2477
3o2i
3565
4109
4652
5196
5740
6283
6826
7370
79' 3
8456
8999
9542
oo85
0628
1171
1714
2257
3342
3885
4427
'^.
6o54
6596
7i38
7680
8222
6
94S
♦oo4
7 8 9
1094
1640
2i85
373©
3275
3820
4365
8x78
8723
8233
8777
92679321
" 9866
9811
o355
0900
1444
1088
2532
3076
36i
4i6
4707
525o
o4xo
0954
1498
2042
2586
3x3o
013674
34217
4761
53o5
5794
6337
6881
7424
7967
85ix
9054
9597
ox4o
o683
1226
1768
23X1
2854
3396
3939
448i
5o24
5566
6108
7735
8277
8764;88i9
930693609415
9848
0390
0931
1473
9902
0444
oo85
1327
95x3
oo58
1 149
1694
2239
2785
333o
3875
4420
5848
63o2
7478
8022
8565
9108
9651
oiq4
0737
X280
X823
2365
2908
345i
3
5078
5620
6i63
665o67o5
7193
7247
7789
833 X
8873
9956
0498
io4o
i58i
9567
oxi3
o658
1203
1749
2294
28^9
3384
3929
4474
5oi9|5o74
5564
6109
6653
7198
7743
8287
883 1
93769430
9920
o464
1008
i552
2096
2640
3x84
3728
4272
48x5
3239
3782
43a6
4870
5359|54i3
5903
6446
8076
9705
0348
079X
i334
1877
3430
3962
35o5
4o47
4590
5x32
5674
6217
73ox
7843
8385
8927
♦0x0
o552
9622
0x67
07x3
1258
i8o3
2349
2804
3439
DiC
56x8
6x63
6708
725a
7797
834i
8886
9974
o5i9
1 06 J
1607
2l5l
2695
5
6
7044
7587
8i3o
86198^4
9x62
9217
9760
o3o3
08^
x388
1931
2474
3oi7
3559
4t02
4644
5x86
5729
6271
67596813
" 7355
7807
8439
8981
9469 9523
oo65
0606
1x48
1690
01234 56789
Diff.
DifllA&Mah.
X
a
3
4
5
6
7
8
9
I
I
3
3
4
5
6
7
8
9
55
no
i65
aao
275
33o
385
440
495
54
108
i6a
216
270
324
378
432
486
DiflllbMolL
L06ABITHM8 OF NUMBSBS.
121
123456789
904i744
3385
2837
3368
170811853
3339
3881
3433
39093963
445o 45054559461 3
5533
6074
66i5
8337
8778
9318 9373 9436 9480
98599913
90503990453
09400904
i48o 1 534
ao3o|3074
361 5
3i55
36o5
4335
5360 53t4 5368
58oo 5854 5908
634o 6394 6448
688069
7419747317
^9577
906 01 16
o655
1734
3373
3813
6044
6583
7I3I
7659
8197
8735
9373
• 9813
0887
i4a5
iq63
3$OI
do38
3576
5o46 5io6|5i54
5587 5641
61386183
6669 6733 6777
7310
775o
8391
8833
477548394883
801 3
8553
9003
963i
0170
0700
1 348
1788
3337
3865
335o3464
38893943
44284483
5o9o
5559
6098
6636
7174
77' 3
835i
9337
9865
b35o|o4o3
0941
3393
3935
3476
4017
7364 7318
7804
8345
8886
9967
o5o7
ro48
i588
3138
3669 3733
3309 3363
37493803
43894343
8067
8606
91469
9685
0334
0763
i3o3
i84i
338o
3919
6690
7338
7767
83o5
87898843
938i
99191
0457
0995
1479 '^33
3017 3071
5553608
i|3i46
3684
1906
4072
83998453
♦03I
o56i
1 103
i64a
3l83
8l31
8660
0817
1 356
1895
3434
3973
35i3
4o5i
5i38
56^7
6i5i|69o5
6744
7383
7830
8359
8897
9455
9973
o5ii
1049
i587
3134
3663
3300
3737
i960
35oa
3o43
3584
4i36
4667
5308
5749
6390
683 1
7373
79'3
0075
061 5
ii56
'^
3336
3777
3317
3857
4307
4937
5476
6016
6556
7006
7635
0871
i4io
3488
3o37
3566
4io5
45904643
5i83
5731
8951
9489
♦037
o565
iio3
i64o
3178
3716
3354
379'
3ui4|3o69
4721
5363
58o3
6344
6885
7436 7480 7534
9048
95899643
0935
1464
3003
3542
3o8i
4775483948834937
8o3i
856i
9103
01390183
06690734
I3IO 1364
1750 i8o4
33903344
383i
3371
3911
445i
499>
553o 5584 5638
6070 6134 6178
66106664 6718
714972037357
768977437797
8383
8833
9361
9901
2o57
3596
3i35
2133 2177
3556 2610 3664 3718
3097 3i5i 3306 3360
3639 3693 3747 38oi
4i8o 42^44388 4343
64536
8075
86i5
9x56
9697
0337
0778
z3i8
x858
3398
3939
3479
4019
455946
504550995
8336
8876
941 5
9954
044004940548
7588
8139
8670
9310
975 1
io33 1087
1573 1626
2riJ 3i65
365o 3704
31893343
3630 3674
4i594ai2
4697 4751
5336 5390
57745838
6367
6905
7444
7982
8520
90049058
9543 " ^
770 3833
3899I395
9113
"i5o
0188
0736
1364
17481 180a
334o
3877
34i5
48594913
6474 6538
8(
857418638
9166 9330
97049758
>343 0396
>78o98j4
1909
2447
3o85
3533
4060
mtL Dur. «t M«K.
N». 01234 56789
54
9 486
DilR
I
y 5,3
a
106
3
159
4
313
5
365
6
3i8
7
371
8
424
9
4n
Dift*;lf«ll.
123
LOGARITHMS OP HUKBSRS.
Nam.
8080
8i
82
83
84
85
86
87
88
89
8090
9«
9^
t
p
99
8100
01
03
o3
o4
o5
06
07
08
09
81 10
II
la
i3
i4
i5
16
17
18
»9
81 ao
ai
aa
a3
24
aS
a6
a?
a8
81 3o
3i
3a
33
34
35
36
3?
38
39
0123 4-56789
9074114
465 1
4167
4705
5i88|5a43
5726,5780
6a63,63i7
6800
7337
7874
84m
8948
9485
9080022
0559
i63
ai6o
a7o5
3241
3778
43i4
485o
5386
5922
6458
6994
753o
8066
8602
9137
• 9673
9090209
0744
1279
i8i5
a35o
a885
3490
3955
4490
5o25
556o
)5
o
7i65
7699
8234
876ft
93o3
• 9637
910 0371
0905
i44o
1 074
a5o8
3o42
68546908696
7391
7928
8465
9002
9539
0076
0612
1149
1686
2222
2759
3295
383 1
4368
4904
5440
5776
65i2
7048
7584
8120
8655
9191
9727
oa62
0798
1 333
1868
24o4
2939
347?
4009
4544
7445
7982
'%
9056
9593
0129
0666
507951
56i4
6149
6683
7218
7753
422114275
4759I4812
52061 536o
5833
6370
5887
64a4
74o8
8o36
8573
9646
oi83
0720
i2o3 ia56
1739 1793
2276
2812
4329
4866
54o3
5941
647«
701 5
755a
8089
86a6
9109 9163
2329
2866
3349 3402 3456
3885i393o3<
9700
0237
0773
i3io
1847
2383
2920
44a I
4957
5404
6o3o
6566
7102
8173
8700
9245
447!
5oii
5547
\t^
5o65
56oi
6ob3!6i37
66191
7i55
7691
8227
6673
7209
7745
8280
9208
9780^98^4
o3i6
o85i
i386
1922
2457
4062
5667
6202
6737
7271
7806
8762I8816
o369
0905
i44o
1075
25ll
9352
9887
o423
0958
1494
2029
2564
3o46
358 1
4ii
465 1
5i
64
86 5239
5721
6256
67906844
7325
7860
8287 834i 8394
882 2I8875 8920
9356'94o9'946j
9890I9944I9997
042 5 0478 o532
0950
1495
2027
256i
1012
ao8i
26(5
3095 3i48
1066
i546ii6oo
35763629
4100
464J
5i77
6710
4i63
4607
523o
5764
2i34
2668
3 202
3736
4a7o
3682
4ai6
4750 48o3
5a84l5337
58x715871
3634
169
4704
5774
63o9
7378
7913
8447
8982
9516
♦o5i
o585
ITIO
i653
2187
2721
3255
4857
5390
5924
60486102
4382 4436 4490
4920 4974 5027
54575511 5565
65^6585
70697123
76007660
8143 8197
86808734
9217 9270
975498079861
0398
0934
1471
aoo8
0290
0827
i364
1900
a437
35io
4a46
4582
5ri8
5654
6190
7262
7798
134
8334
94o5
9941
0476
1012
1 547
2082
a6i8
3i53
3688
4223
4758
5293
58a8
6362
6807
7432
7966
85oi
9035
o344
0881
i4i7
1954
3490
3o27
3563
5172
5708
6244
66396693
7176
77t3
825o
8787
4544
5o8i
56i8
6i56
11^
83o4
884i
93a4 9378
9915
045 1
0988
1525
ao6i
2544 a5o8
3o8o3i34
36173670
41534207
46894743
5225
4
567a
6209
6747
7a3o|7a84
7821
5762
6296
672667806834
731673697433
785a
8387
8870 8923
"9459
9994
o53o
io65
1601
2i36
2671
3206
3741
4276
48ii
5346
588i
6951
7485
8554
9089
95709623
oi58
0692
2294
2828
0104
o638
1 173
1707
aa4i
2775;
3309*3362
3842I38C6
43764430
4910*4963
5444|5407
5977l6o3i
7905
844 1
8977
9312
♦o48lbioi
o583
1119
i654
2189
2725
3260
3795
433o
4865
5400
5935
641664696523
7004
753o
80208073
8608
9142
9677
021 1
0745
Taa6fa7Q
1760 i8tJ
a348
a882
34i5
8358
8895
943a
9068
o5o5
X042
1578
aiiS
265i
3i88
3724
4a6o
4797
5279 5333
58i5
635 1
6887
9o3o
956696
06370690
1226
1172
1708
2243
2778
33i3
3848
4383
49«8
5453
5o88
5869
64o5
6941
7477
801a
8548
9?84
61 55
7o58
7592
8127
8661
9106
973o
0264
0799
1313
1867
2401
2935
346d
3949I40034056
4483 45364599
50175070
555o
6084
1761
2207
28Ja
3367
390a
4437
497a
5507
6o4a
6576
711 1
7646
8180
8715
9249
9784
o3i8
o85a
i386
1920
2454
2988
3522
5i23
56045657
61376191
DiC
01984 56789
vm,i
I Dur.
I
a
3
4
5
6
I
S4
108
16a
ai6
370
3a4
378
43a
486
a
3
4
5
6
7
8
53
106
159
212
a65
3i8
371
4a4
477
Diir.4b
k.
LOaARITHMS OP NUMBEES.
123
8140
4i
4a
43
M
45
46
47
48
49
8i5o
5i
53
53
54
55
56
57
58
59
8i6o
6i
6a
63
64
65
66
68
69
8170
71
7a
73
74
75
76
77
78
79
8180
81
8a
83
84
85
86
87
88
89
8190
9»
93
94
99
12 8 4
5 6 7 8 9
010 6a44
6778
73ii
7844
8378
8911
9444
• 9977
911 o5io
1043
1576
aio9
a64a
3i74
3707
4240
477a
53o5
5837
4293
4825
5358
5890
6369 64a3
690a
7434
7966
8408
9o3o
• 956a
9120094
o6a6
ii57
1689
aaai
a75a
3a84
38i5
4346
4878
4931
5409546
5946
6471
700a
7533
8064
8595
913 0187
07170770
ia48 i3oi
1778 i83i
3309 a36a
38393893
33693433
4960
5490
6549
7079
6397
683 1
7364
7808
843i
89649018
955 1
0084
17
ii5o
♦o
o5
1096
o564o6
6a9|i683
aai5
3748
3281
3760I3814
I
ai6a
2695
3a 38
69557008
7487
8019
855 1
9083
9615
of47
0679
laio
1743
3374
38o5
3337
3868
65^
7055
7586
81 17
8648
9136 9179 933
96569709976
0340 0393
5oi3
5543
660a
71 33
7609I7663
635i
6884
7418
7951
8484
4346
4879
5411
5943
6476
7540
6073
8604
91 36
9668
O30O
0733
1364
1795
a3a7
3858
3390
3921
43994453
4o84
55i5
6o46
65a46577
7108
8170
8701
0834
1 354
1884
a4i5
1945
3475
4536
5o66
5596 564q
60191607361356178
6655
7185
64o4i6457
6938
747 »
8004
8538
9071
9604
oi37
0670
I303
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LOGARITHMS OF KVMBBBB.
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127
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LOGARITHMS OF NUMBERS
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8440
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43
44
45
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47
48
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LOGARITHMS OF NUMBERS.
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2000
25lO
30193070
3529 358o
4549
SoSo
5568
7555
8064
8574
9083 9134
9592
OIOI
0610
9643
01 52
0661
1170
1679
2188
2697
32o5
3714
4223
4680 473 1
4783
5189 5240 5291
5697 5748 5799
6206 6257 63o7
6714 6765 6816
7373
7781
.289
8798
1 337
1845
2352
2860
3368
3875
438a
1 540
ao5i
356i
4600
5iio
5619
7i48
7657
8166
8676
9185
9694
0303
0713
isai
1730
a33o
3748
3256
3765
4^4
7334
7833
834o
8848
93669356
98149864
o32i 0372
08290880
" i388
1896
24o3
2911
3418
3926
4433
4445 4496 454714598 4649
4956 5oo7i5c>58;5i09 5i6o
5466 5517I 5569' 56ao
5977!6o28!6o79 61 3o
6488 6539 6^90
6098
7509
7o5o
7560
80208071
853o
9040
858i
9091
9551 9602
0112
0061
0571
1081
1 591
2ioa
261a
3l2I
363i
4i4i
465i
5i6o
5670
6i8o|623i
66896740
7*9?
7708
8217
8737
9336 9287
9745
0354
0763
137a
1781
3390
3798
33o7
38i6
4334
4833
5341
585o
6358
6867
7375
7883
8391
88998960
9407
0423
0931
1439
1946
3454
3963
3469
39774
4484
0623
Il33
1643
2i53
2663
3172
3682
4192
4702
5aii
5721
7250
7759
8268
8777
7101
761 1
812a
863a
914a
9653
oi63
0673
ii83
1694
2204
2713
3323
3733
4343
4753
5a6a
577a
6a8a
6791
73oo
7810
83i9
8838
9338
9847
o356
o865
1 374
1883
3391
3900
3400
3918
44a6
4935
5443
5962
64o9|646o
69176968
74a6
7934
8443
8960
9458
9966
0474
098a
1489
1997
35o5
3oi3
3530
037
4535
o3o5
0814
1 323
1 83a
3341
a84o
3358
3867
4375
4884
539a
5901
7476
7934 7985
♦017
o5a5
io3a
1 540
3o48
3555
3o63
3571
4078
4585
5671
6181
6693
73o3
77i3
83a4
8734
9345
97049755
0265
664 1
71 5a
766a
8173
8683
9194
oai4
0734
ia34
1745
3355
3764
3374
3784
4394
4804
53i3
58a3
6333
684a
735 1
7861
8370
8870
9388
9898
0407
0916
i4a5
1933
3443
3961
3460
3968
4477
8493
9001
9509956096
0067
0675
io83
1 591
3099
3606
3ii4
3631
4139
4636
0775
1385
1796
33o6
a8i5
33a5
3835
4345
4855
5364
5874
6383
6893
74o3
7913
8431
8930
9439
^52
0967
1475
1984
3493
3003
35ii
4019
4538
4986 5o36
5494" '
600a
65ii
7019
7537
8o35
8544
9062
5545
6o53
656a
7070
7578
8086
8594
9103
'10
0118
0636
ii34
i64a
3 149
3657
3i65
3673
4180
4687
12 3 4 5 6 7 8 9
17 ~"
Dir.
5i
Diff
DtfL^MoH.
I
3
3
4
5
6
7
8
I
3
3
4
5
6
7
8
9
5i
103
1 53
3o4
355
3o6
357
4o8
459
5o
100
i5o
300
35o
3oo
35o
4bo
45o
DICItMolL
190
LOQABITHMS OF NUMBERS.
Nub.
SS60
6i
6a
63
64
65
66
68
69
8570
71
7a
73
74
75
76
77
78
79
85 80
81
8a
83
84
85
86
B7
88
«9
8590
9«
9a
98
99
8600
01
oa
o3
04
o5
06
07
08
09
8610
IX
la
i3
14
i5
x6
X7
18
19
Nan.
12 3 4
•39 4738
5a45
575a
6a 59
6767
7274
7781
8a8d
8795
9301
9808985099109960
o3i5o366 0416 0467
933b3i5
08a a
i3a8
1 835
a34i
a848
3354
386o
4367
4873
5379
5885
6391
689769486998
• 993a
9340437
i448
1953
2459
2964
3469
3974
. 4479
4985
5489
5994
6499
7004
1^
85i8
9oa3
95a7
935oo3a
o536
io4o
i544
ao49
a553
3o57
356i
4o65
4788
5a96
58o3
63io
6817
73a4
783 1
8338
8845
935 a
087a
1379
i885
a 39a
a898
34o5
391 1
4417
49a3
543o
5936
644a
2
74o3
89ao
94a6 9477
998a
o488
7454
7951
846
8971
09430993
1499
aoo4
a5o9
3oi5
35ao
4oa5
453o
5o35
5540
6045
655o
7054
7559
8064
8568
9073
9578
008a
o586
1091
1595
ao99
a6o3
3i07
36ii
4ix5
45694619467047
48394890.4941
5346 5397 5448
5854 5904,5955
636i 64ia'646a
686869196969
7375 74a6 7476
788a 7933 7983
838984408490
8890 8947 8097
94o3 9453 9504
♦on
64i66467|o5i8
0974 ioa4
09a3
i43o
1936
a443
2949
345:>
396a
4468
4974
5480
5986
6492
7504
8010
85i6
90a I
95a7
♦o33
o538
io44
1 549
3o55
a56o
3o65
3570
4075
4580
5o86
5590
6095
6600
7io5
7610
81 14
8610
91 a3
96a8
01 3a
o637
n4i
i645
ai5o
a654
3i58
366a
4166
i48o
1987
i53i
ao37
a493 a544
3ooo
35o6
4oia
4519
5oa5
553 1
6o37
6543
7049
7555
8061
8566
907a
9578
oo83
0589
1094
1600
aio5
a6io
3ii6
36a I
4ia6
463i
9174
9678
oi83
0687
1 191
1696
aaoo
2704
3ao8
371a
4ai6
ao
3o5o
3557
4o63
4569
5075
558i
6088
6594
7099
7605
8111
8617
91 a3
96a8
01 34
0639
ii45
i65o
ax56
a66x
3i66
3671
4176
468a
5x36
564i
61466196
665i -
7x55
5x87
5691
6701
7206
7660 7711
8x65 8ai5
86698730
9aa4
9729
oa33
0738
xa4a
1746
aa5o
2754
3a59
3763
4a66
4770
499«
6006
65x3
7oao
75a7
8o34
854i
9048
9555
006a
o568
1075
1 58a
ao88
a595
3ioi
36o7
4xi4
46ao 4670
5xa6
563a
6x38
6644
7i5o
7656
816a
8668
9173
96799729
0184
0690
1195
1701
2206
2711
3ax7
37aa
4aa7
4732
7761
8770
9375
a8o5
3309
38i3
43x7
48ai
678
5o4a 5093 5x44
5549
6057
6564
7071
7578
8o85
859a
oiia
06x9
1x20
x632
ai39
a645
3i52
3658
4i64
5x77
5683
6x89
6695
7201
7707
8aia
87x8
9224
oa35
0740
xa46
i75x
aa57
a76a
3a67
3772
4277
4783
5237
5742
6a47
675a
7a56|73o7
5a87
5792
6297
680a
781a
8a66 83x6 8367
786a
8367
887 X
937G
9779I9830 9880
88a I
93a5
oa84o334
0788
129a
1797
a3ox
o838
1343
x847
a35i
a855
3359
3863
4367
4871
566o
6107
66x4
7xaa
7639
8x36
8643
&
oi63
0670
1176
1 683
2x89
2696
3ao2
3700
42x5
472 X
5a 27
5733
6239
6745
7a5x
7757
8263
8769
9^74
9780
oa86
079X
1396
x8oa
a3o7
a8i2
33x8
3823
4338
4833
5338
5843
6348
6853
7357
o385
0880
139S
1897
2402
2906
34x0
3914
44x8
4922
565 1
6x58
6665
7172
7679
8x86
8693
9200
9707
0214
0720
1227
1733
2240
2746
3253
3759
4a65
4772
5378
5784
6390
6796
73o3
7808
83x3
88x9
9335
983x
o336
0843
x347
1 853
3358
3863
3368
3873
4378
4884
5388
5893
6398
6903
74o8
79x3
8417
8933
9376 9436 9477
9931
0435
0939
1444
1948
345a
3956
3460
3964
4468
4272
5x^
5703
6300
6716
7333
773o
8337
8744
935 X
9758
0364
0771
1378
1784
239X
2797
33o3
38io
43i6
4833
5338
5834
634x
6846
7353
7858
8364
8870
9375
988X
o387
0893
X398
1903
3408
29x4
3419
3934
44a9
4934
5439
5944
6449
6954
7458
7963
8468
8973
9981
0485
0990
x494
X098
35o3
3oo6
35ix
4oi5
45x8
5o33
01234 56789
DtC
Dili:
DiiRJtlfall.
I
3
3
4
5
6
7
8
9
5i
I03
1 53
204
355
3o6
357
408
459
I
a
3
4
5
6
7
8
9
5o
xoo
i5o
aoo
aSo
3oo
35o
4oo
45o
Diff.
LOGARITHMS OF NUMBERS.
131
Ni
8630
ai
22
23
24
25
a6
27
2&
39
86 3o
3i
32
33
34
35
36
37
38
39
86 4o
4i
42
43
44
45
46
47
48
49
86 5o
5i
5a
53
54
55
56
57
58
59
8660
61
6a
63
64
65
66
err
68
69
8670
71
72
73
74
75
76
77
78
79
Ffftm*
12 3 4
93ft 5o73|5ia3
5576
6080
6584
7087
7591
8095
8598
9101
• 9605
9360108
061 1
iii4
1617
aiao
a6a3
3i36
3629
4i3a
4635
5i37
564o
6143
6645
7148
7650
8i52
8655
9157
• 9659
987 0161
o663
ii65
1667
2169
2671
3172
3674
4i
4677
764
568o
6i8a
6683
8187
8688
• 96909740
9380191
069a
1193
1693
2194
2696
3195
3696
4697
5627
6i3i
6634
71 38
7t>4i
8145
8648
91 52
96559705
01 58
0661
ii65
1668
ai7i
a674
3l77
2724
3227
367913730
' 4a33
4735
4182
4685
5i88
5690
6193
6695
7198
7700
8ao3
8705
9207
9709
oaix
0713
iai5
1717
aai9
a72i
32a3
3724
226
4728
5731
6232
^33
71847235
76867736
8237
8738
0241
0742
1243
1744
2244
a745
3a45
3746
3796
4196 4a47 4397
4747
5173
5677
6181
6685
7188
7692
8195
5224 6274
5738 5778
623 1 6282
6735 6785
7239 7289
9202
774a
8246
86998749
925 a
9756
020903590309
0712
I2l5
1718
2221
5238
5741
6243
6746
7248
775o
8a53
8755
7801
83o3
88o5
93579307
97599810
oa6i o3xa
O764|o8]4
t3i6
1818
19
ia65
1767
226923
2771
3273
3775
4276
4778
5179 5229 5279 5329
7786
8287
8788
9189 9239 9289 9339 9389
9790
0291
0792
1393
1794
3394
3795
4797
076a
ia65
1768
3371
2774
3377
3780
4383
4786
5388
5791
6293
6796
7398
3831
33a3
38a5
43a6
4838
583 1
6333
5781
6383
6783
7a85 7335
7836
8337
8838
9840
o34i
0843
1 343
1 844
334^
3845
3396 3346
3846
4347
4J47
7793
8396
9806
o8ia
i3i6
1819
3333
3825
3337
383o
4333
4836
5338
584i
6344
6846
7349
785i
8353
8855
9358
9860
o362
0864
i366
1868
2370
2871
3373
3875
4376
4878
538o
588i
6382
68346884
7385
7886
8387
8888
9890
0391
0892
1393
1894
a394
3895
3396
3896
4397
4897
5 6 7 8 9
5335 5375i5435i5476
5838
6333
6836
73397390
7843
83.^
885o
9353
9856
o36o
o863
i366
1869
3373
3875
3378
388i
4383
4886
5389
5891
6394
5439
594a
6444
68966947
73997449
7901
84o3
89068956
9408
9910
o4ia
0914
i4i6
1918
a4ao
3933
3433
3935
44a7
4938
543o
5931
6433
7435
8437
9440
9941
o44i
094a
1443
1944
a445
3945
3446
3946
4447
4947
5879593915979
638ai6433 6483
6886:6936 6987
7893
83978447
8900I895
94o3
99^7
o4io
0913
i4i6
1919
3433
3935
3438
3931
4434
49364987
7951
8454
5489
5993
6494
6997 _
7499 7550
8o53
8554
9458
9960
0463
0964
1466
1968
3470
3973
3474
3975
4477
4978
5480
5981
6483
69346964
7485
79367986
8488
8939 8989 9039
9490
9991
049a
0993
1493
'994
3495
3995
3496
4497
4997
7440
7943
o
9454
9957
o46o
0963
i466
1970
a473
3975
3478
3981
4484
8ooa
85o4
90069056
9508
♦010
o5i3
ioi5
i5i6
aoi8
3530
3033
3534
4o35
4537
5o38
553o
6o3i
6533
7034
7535
8037
8538
9540
♦04 1
o54a
1043
i543
3044
3545
3o45
3546
3996 4o46 4096
4547
5o47
55a6
6o3o
6533
7037
7541
8044
8548
905 1
9554
oo58
o56i
1064
1 567
ao70
3573
3076
3579
4083
4534 4584
7490
8497
9001
95o4
♦007
o5ii
ioi4
i5i7
3030
3533
3o36
3539
4o3i
5o37
5540
6043
6545
7047
95599609
0061
o563
io65
i567
3069
3570
3073
3574
4075
4577
5079
558o
6083
6583
7084
7585
8087
8588
9089
9590
0091
0593
1093
1593
3094
3595
3095
3596
4597
5097
5087
5590
6093
6595
7097
7600
8103
8604
9107
01 1 1
o6i3
iii5
1617
3119
3631
3l33
3634
4136
46a7
5139
563o
6i33
6633
7134
7636
8137
8638
9640
oi4i
0643
1143
1643
ai44
3645
3i45
3646
4i46
4647
5i47
01234 56789
Biff.
Diff.
DUr. k. Mult.
I
2
3
4
5
6
7
8
9
5i
103
i53
3o4
a55
3o6
357
408
459
I
5o
2
100
3
i5o
4
200
5
25o
6
,V)o
7
35o
8
400
9
45o
Diff. «B Mult.
ia2
LOGARITHMS OF NUMBERS
Nora.
8680
8i
8a
83
84
85
86
«7
88
89
8690
9'
9a
93
94
i
99
8700
01
02
o3
04
o5
06
07
08
09
87 10
II
12
i3
i4
i5
16
17
18
»9
8730
ai
aa
a3
a4
a5
a6
II
39
8730
3i
3a
33
34
35
36
37
38
39
Num.
12 3 4
5 6 7 8 9
938 519-^5247 5397,5347.5397
5698157485798
6198 6a48>6298
6698
7198
7698
8198
8698
9198
• 9698
9390198
0697
1197
1697
2196
3696
3195
3695
4194
4693
5193
5692
6191
6600
7189
7688
8187
8685
9184
• 9683
9400182
0680
1179
1677
2176
2674
3172
3670
4169
4667
5i65
5663
6161
6748
7248
7748
8248
8748
9^48
9748
0248
0747
1247
1747
2246
2746
3245
3745
4244
4743
6659 6709
71 5? 7206
7157
7654
8i52
865o
9147
• 9645
94x 0142
0640
II 37
1635
2l32
a6a9
Biao
36a3
'4t20
4617
5242
5742
6241
6740
7239
7738
8237
8735
9234
9733
oa3i
073©
1229
1727
2225
2724
3222
3720
4218
4717
52i5
5713
621 1
72(
7704
8202
8700
9695
0192
0690
1 187
1684
2182
2679
3176
3673
4170
4667
6798
7298
7798
8298
8798
9298
9798
0298
0797
1297
1797
2296
2796
3295
3795
4294
4793 4843
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0630
1118
3094
368a
3070
3668
4o46
4634
6033
65io
6900 6949 6998 6047
746i
7948
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94"
5
0873
x36o
i847
3334
3631
33o8
7473
9164 9313
9701
0190
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1 167
1606 1666
a 143
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3119
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3
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6669
7033
7610
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9363
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1944
343i
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3406
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6121
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1364
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224 r
2729
3217
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4193
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7607
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8682
9069
9608 9557 9606
0093
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1019
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1 362
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33 14
38o2
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5266
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1 654
2042
2629
3oi6
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7706
8192
8680
9167
9654
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0629
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2090
2677
3o64
355i
6268
6768
6248
6/38
77x8
8208
498608
369187
9677
0166
0666
ii45
i636
2134
2613
3 102
3692
4o8i
4570
6069
5648
6o37
6626
7014
75o3
848o
8969
9457
9946
0434
0923
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2876
3363
386i
4339
4827
53i5
58o3
6290
67396778
7266
7753
8241
8728
9216
97o3
0190
0678
ii65
1662
2626
3ii3
36oo
DUr.
01234 56789
49
DUE lb Malt.
I
3
3
4
6
6
7
8
I
3
3
4
5
6
7
8
9
Difll
1 47
196
245
394
343
393
44i
48
96
i44
19a
34o
388
336
384
433
DiCfcliah.
136
LOGARITHMS OF NUBCBBES.
Nuna.
8920
ai
a3
25
26
27
28
29
80 3o
3i
32
33
34
35
36
37
38
39
89 40
4i
42
43
M
45
46
47
48
49
89 5o
5i
5a
53
54
55
« 56
57
58
59
8960
61
6a
63
64
65
66
67
68
69
8970
71
72
73
74
75
76
77
78
79
1 2
950 3649
4i35
462 a
5109
5596
606a
6569
7o55
754a
8oa8
85i5
90UI
9487
• 9973
951 0459
0946
i43a
1918
2404
2889
3375
386i
4347
4832
53i8l5366
58o3
62896337
8764
9a49
96869734
6774
7a6o
7745
823o
8716
9201
9526171
o656
ii4i
1626
am
2595
3o8o
3565
4o49
4534
5oi8
55o3
5987
6472
6956
7440
8409
8893
9377
• 9861
953o345
o8a8
i3ia
1706
2280
4
4671
5i58
5644
6i3i
6617
7104
7590
8077
8563
9o5o
9536
♦oaa
o5o8
585a
68a3
7308
7794
3746
4233
4720
5ao6
5693
6180
tXXX)
71 53
7639
8126
8612
%
0071
0557
1043
1539
201 5
25oi
2987
347a
3958
4444
492?
54i5
5901
6386
6871
7357
784a
8279 8327
88i3
02190268
07040753
1189
1674
ai59
a644
ia38
i7a3
a 208
3692
3ia93i77
36i3
4008
4582
5067
555 1
6o36
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366a
4i46
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5ii5
56oo
6084
6569
7973
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85o5
8457
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0393
0877
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1 844
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09a5
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1893
3376
1091
1 577
ao63
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3o35
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4007
4492
4978
5464
643?64?3
643
6930
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7891
8376
8861
9346
9831
o3i6
0801
ia86
1771
aa56
2741
3226
3710
4195
70047053
7489753775867634
5i64
5648
61 33
6617
7101
8070
8554
89809038
047^ 0522
9522
♦006
0490
0974
1457
I94I
2425
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1626
2II2
2598
3o84
3569
4o55
4541
5027
55ia
o365
o85o
1 335
1820
23o5
2789
3274
3750
4a43
46794728
5212
5607
6181
6665
7i5o
8118
8602
9086
9570
oo54
o538
1022
i5o6
1980
2473
5
38933941
4379 4427
6 7 8 9
4866
5352
5839
6326
6812
77S
8271
8758
92449293
973o
0216
0703
1189
1675
a 161
2646
3i32
36i8
4io4
4589
5075
556i
6o46
6532
7017
75o2
7988
8473
8958
9443
9928
o4i3
089B
1 383
1868
2353
2838
3322
38o7
4292
4776
5261
5745
623o
6714
71Q8
7682
8167
865 1
9135
9610
01 o3
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1070
1554
2o38
2522
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39894038
44764525
491449^5012
5401 545o 54
5888 5936
6374 643316472
6861
7347
7834
8320
8806
I
977
026
0751
1237
1723
2209
2695
3i8i
3667
4i52
4638
5i24
5609
095
658
7066
755 1
8o36
8521
9007
9492
9977
0462
0^7
i432
1917
a4oi
2886
3371
3856
4340
4825
5309
5794
6278
6762
7247
773 1
8215
9i83
9667
oi5i
o635
1119
i6o3
2086
2570
6958
7445
7931
83698417
8855
9341
9827
o3i4
0800
1286
1772
2258
2744
3229|3
7114
7
8570
9055
9540
♦025
o5io
5842
6326
6811
7295
7779
8263
8747
9231
9715
0199
o683
ii«7
i65i
2i35
2618
8004
9390
9876
o36a
0848
1 334
1820
23o6
2792
278
3764
425o4
4735
5221
5706
3715
4201
4687
5172
5658
61436192
66206677
7i63
7648
8i33
8619
9104
4087
4574
5o6o
5547
6o34
6520
7007
7493
7980
8466
8952
9439
9925
o4ii
0897
1 383
1869
2355
a84i
3327
38i2
298
4784
5269
5755
6240
6726
7211
76J7
8182
8667
91 52
95899637
0074
0559
1044
1529
aoi4
3468
395a
4437
4922
5406
5890
6375
68
734
7828
83i2
8706
9280
9764
0248
073a
iai5
a667
oraa
0607
1092
1 577
ao62
2547
3o32
35i6
4001
4486
4970
5454
6^?
7CJ92
7876
836o
8844
9328
9812
0296
0780
1264
1748
223 1
2715
nmu 01284 56789
Diff.
: lb Molt.
DiflL
I
2
3
4
5
6
7
8
9
I
a
3
4
5
6
7
8
9
%
i47
196
245
294
343
392
44i
4B
96
i44
240
288
336
384
432
DiC
*'''^l
r'
LOOAKITHXS OF N UM B £ B S .
187
iruA
8i
83
83
84
85
86
»7
88
89
8990
9«
99
9000
01
oa
o3
04
o5
06
07
08
09
90x0
II
13
i3
14
i5
16
17
18
'9
9090
ai
aa
a3
a4
a5
a6
a?
a8
29
90 3o
3i
3a
33
34
35
36
3?
38
39
12 3 4
5 6 7 8 9
3763
3a47
3731
431
4697
5]8i
5664
6147
663i
7114
7597
8080
8563
a8i 3 3860 3908 3967
3395(3344 3393'344o
3779383738763934
4436a;43ii 4359*4407
474614794 484a|489'
5339 5a77 5336 5374
57135761 58095857
6196,6344 639316341
' "68a4
9539
9540013
0494
0977
i46o
1943
a435
3390
3873
4355
4837
53i9
58o3
6384
6679
7163
7645
8138
8611
90469094
9577
0060I0108
o543
ioa5
i5o8
1991
67376776
73107359
7743
8335
8708
9191
0591
1074
i556
3
3473 3533
3956'3oo4
585o
6333
67666814
7248
773o
8313
8694
9176
• 96579705
955 01 3910187
06310669
II03 ii5o
i584i633
ao65
2547
3oa8
35io
3991
4472
4953
5434
^
6397
3438
3931
44o3
3487
39694
445i
488549344983
5368 5416 5464
58985946
638o6438
68636910
7396
7778
8360
9743
9334
7694
8177
8660
9143
96359674
oi57
0639
ii3a
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0393087
3570
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3535
017
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1304 I353'l300
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7455 75o3 755 1
7935 7984 8o33
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1 348' 1 396
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0784
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7055
61396187
8453
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1 343
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6x56i6ao4
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1873
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7
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4x66
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4a58
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1931
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1 536
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a45i a4o9
a884 a93a 3980
3365'34i3 346i
384638953943
4338 4376 44a4
480948574905
5390 5338 5386
5771,58195867
6a53!63oo'6348
6733 6781
7314
6839
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83348373
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9137 9185 9334
76477^595
8176
86098657
7363
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XOX3
1493
0x47
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96669714
0195
0675
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Dir.
12 8 4 5 6 7 6 9 DiiT.
18 L*
Diir. it, Molt.
I
a
3
4
5
6
7
8
9
I
a
3
4
5
6
7
8
9
$
147
196
a45
394
343
393
44i
48
96
144
19a
a4o
388
336
384
433
Diff. * Moh.
138
LOGAEITHMS OF NUMBERS.
N«. 12 3
•04O
4i
4a
43
44
45
46
47
48
49
00 5o
5i
53
53
54
55
56
57
58
59
00 6o
6i
6a
63
64
65
66
67
68
69
9070
71
7a
73
74
75
76
77
78
79
9080
81
82
83
84
85
86
87
88
89
9090
9»
s
98
99
Noi.
9561684
9i65
3645
'3i25
36o6
4086
4566 46144663
5o46
5526
6006
6486
6966
7445
7925
84o5
8885
9364
• 9844
957 o323
o8o3
1383
1761
2341
3730
3199
3678
41 57
4636
5ii5
5594
6073
6553
7o3o
75oo
7988
8466
8945
9433
958o38o
o858
1 337
i8i5
3393
2771
3a49
3737
43o5
4683
f0 5i6i
5639
6117
6594
7073
50945143
5574 5633
6o54 6103
6534,6583
70147063
74937541
79738021
8453 85oi
8933.8980
94139460
98929940
0371
o85i
8037
85o5
8983
• 9937
1733 1780
33l3 3361
2693 2741
31733221
3654 3702
4i344i82
i33o
1809
2389
3768
3347
0419
0898
1378
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3336
3816
3395
3736I3774
43o5 4353
46844733
5i63 53ii
5643
6l3I
6600
7078
7557
8o36
85i4
8993
9471
9950
0438
0006
i385
i863
3341
2819
3397
3775
4a53
473 1
5309
5687
6164
6643
71 30
7549 7597
8075
8553
9o3o
94599507
9984
6647
7126
7605
8o83
8563
9041
9519
0476
0954
1433
191 1
3389
2867
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3833
43oi
4779
5257
5734
6212
6690
7167
7645
8123
8600
9077
4710
5190
5670
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66306678
71 58
7110
7589
8069
85498597
9028
9508
9988
0467
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i4a6
1905
3384
2864
3343
3822
43oi
4780
5359
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6169I6317I6364
6695
7174
7653
8i3i
8610
9088
9567
♦045
o534
I003
x48o
1958
3437
3915
3393
3871
4349
4837
53o4
5783
6360
73i5
7693
8170
8648
9125
9555J9603
♦o33|oo8o
4
1876
3357
3837
33i7
5 6 7 8 9
1838
3309
3789
3369
37503798
43304378
4758
5338
5718
6198
7637
8117
9076
9556
♦o35
o5i5
1474
1953
2433
3391
3870
4349
4838
53o7
5786
6743
7333
7701
8179
8658
91 36
9615
0093
0571
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1 538
3006
3484
3963
3441
4^9^
4874
5353
583o
63o8
67386785
7363
7741
8318
8695
9173
9650
01 38
1935
34o5
3885
1973
3453
3933
3365 341 3
38463894
43364374
4806
4854
5386,5334
5766 5814
63466294
6726 6774 682a 6870
7306 7354
7685
8i65
8645
9134
9604
oo83
o563
io4a
1533
3O0I
3480
3959
3439
3oi8
4^97
4876
5355 54o3
5833
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7370
7748
8337 8375
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9663
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06190667
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1576
2054
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3oio
3488
3966
4444
4933
5400
5878
6355
6833
73ii
7788
8743
9331
9698
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3o6o
35ii
3962
44i3
48194865
5271
5722
6173
65796624
7029I7075
7480,7525
7931
8382
7976
8427
53i6
57^
6218
6669
7120
7571
8021
8472
8743 8788 8833 8878 8023
9193 9238 9283^9328 9374
9644J9689 9734 9779 9824
0095:01400185 0230.0275
tf545'o59o>o635 0680*0725
DiClfclfalt.
I
a
3
4
5
6
7
8
i
01234 56789
DHT
46
t84
a3o
276
332
368
4i4
t
45
3
3
rfs
4
180
5
aa5
6
370
7
3i5
8
36o
9
4o5
Diir.«tM«h.
148
Nttub
9640
4i
4a
43
44
45
46
47
48
49
96 5o
5i
5i
53
54
55
56
57
58
59
9660
61
6a
63
64
65
66
67
63
69
9670
71
7a
73
74
75
76
77
78
79
9680
8t
8a
83
84
85
86
87
88
89
9690
9»
9a
98
99
Nsm*
LOGARITHMS OF NUMBERS.
12 3 4
5 6 7 8 9
IHC
Di&^Mah.
9d4 0770
I67I
aiaa
a57a
3033
3473
39a3
o8i5|oS6o
ia66
1716
2167
3617
i3ii
1761
3212
266a
35 1 8
3968
4373|44i8
4623'4B6&
3067' 3i 1 2
5273
5723
6173
6623
7073
53i8
576H
6218
6668
7118
3563
4ot3
4463
4913
5363 5408
0905
i356
1806
2257
2707
3i57
36o8
4o58
45o8
4958
752317568
7973!8oi8
8422
8872
9322
• 9771
9S50221
0670
1 1 20
1569
8467
9367
9816
oa66
0715
!i65
1614
aoi9 2064
24681 a5 1 3
391712962
3566 3411
38i6
4a65
4714
5163
56i2
6061
65fo
6959
740-'
7856
83o5
8754
9202
• 9651
0996
1445
1893
234i
2790
3238
3636
4i34
4582
5o3o
5478
5q26
386i
43io
4759
5208
5657
6106
5858
63o8
6758
7208
7658
8107
8557
9007
9457
o356
o8o5
12 10; 1 255
1659 1704
2108 2i53
'>Si3
6a63
6713
7i63
t6i3
8<)63
85i2
8962
941a
9861
o3ii
0760
2558
3oo7
3456
3935
4355
48o4
5253
5702
6i5i
6555 6600
70o3
7452
5o
8798
9247
9696
01 44
0593
io4i
1489
io38
2 586
2834
7048
7497
7046
8395
8843
9292
9740
0189
0637
1086
1 534
19S3
243 1
2879
2603
3o52
35oi
3950
4399
4849
5298
5747
6196
6644
7093
7542
799»
844o
8888
9337
9785
0234
068a
Tl3l
1 579
2027
3476
3934
32833337.337a
373113776,3820
4179142244368
462714672,4717
5o75|5iao5i65 5209
0951
i4ox
i85i
23oa
3753
33oa
3653
4io3
4553
5oo3
5453
5903
6353
68o3
7253
7703
8i5a
860a
9o5a
95oa
9951
o4ot
o85o
i3oo
1749
2198
2648
3097
3546
3995
4444
4893
534a
5791
6340
6689
71 38
7587
8o36
8484
8933
938a
9S3o
0279
0737
1176
f6a4
2073
a5ai
3969
3417
3865
43i3
4761
5533
5971
6J74|64i9
683316867
737017314
5568
56»3
6oi66fj6o
6464!65o8
6911 6956
7359! 7404
5657
6io5
6553
7001
7449
i446
1896
a347
2797
3247
3698
4i48
4598
5o48
5498
5948
6398
6848
7298
7748
8197
8647
9097
9546
9?9f
0446
0895
1 345
1794
3343
3693
3i4a
3591
4o4o
4489
4936
5387
5836
6a85
6734
7i83
7633
8081
85298574
8978
94a6
9875
o334
0773
I230
1669
aii7
3565
3oi4
3462
3910
4^58
48()6
5354
5703
6i5o
6598
7046
7493
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1491
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3^93
3842
3393
3743
4193
4643
5093
5543
5993
6443
6893
7343
7793
8342
8693
9»4a
9591
♦o4i
0491
09400985
i58oi434
1389
1839
3388
2737
3187
3636
4o85
4534
4983
5432
5b8i
633o
6779
7228
7677
8125
9033
^71
9030
o368
0817
1365
1714
3163
3610
3o58
3507
3955
44o3
485 r
5299
5747
6195
6643
7090
7538
1086)
1 536
1987
3437
2887
3333
3788
4338
4688
5i38
5588
6o38
6488
6938
7388
7838
8287
ii3i
i58i
3o3?
3483
3932
3383
3833
4383
4733
5i83
1176
1636
ao77
2527
2977
34a8
3878
43a8
4778
5aa8
5633 5678
6o83
6533
6983
7433
7883
8332
0086
o535
873718782
918719232
96369681
oi3i
o58o
io3o
1479
1929
3378
2827
3277
3726
4175
46^4
5073
5533
5971
6430
1434
1884
3333
3783
3333
368i
4i3o
4579
5o38
5477
5926
6375
6824
7373
7722
8170
8619
9068
9516
9965
o4i3
0862
i3io
1758
2307
3655
3io3
355 1
4000
4448
4896
5344
5792
6240
6687
71 35
7583
7766
8215
8664
9112
9561
♦010
o458
0907
1 355
i8o3
2252
2700
3i48
3596
4044
4493
4941
5389
5836
6284
6732
7180
7628
61 a8
6578
7028
7478
7928
8377
8827
9277
9736
0176
0635
1075
1 534
1974
34a3
2873
3331
3771
4a ao
4660
5ii8
5567
6016
6465
68696914
73i8 7363
7811
8360
8709
9157
9606
oo54
o5o3
0951
i4oo
i848
3396
5745
3193
364i
4089
4537
4985
5433
588 1
6329
6777
7225
7673
45
I
a
3
4
5
6
7
8
9
46
i38
i84
a3o
376
3aa
368
4i4
I
a
3
4
5
6
7
8
45
180
aa5
270
3i5
36o
4o5
01234 56789
DUt
DiClbMtBlt.
LOGARITHMS OF HUMBBSS.
14*
9T0O
01
oa
o3
o4
o5
06
07
08
09
9710
II
la
i3
14
i5
16
>7
18
>9
9720
ai
23
M
25
26
27
26
7730
3i
32
33
34
35
36
37
38
39
9740
4i
4a
43
44
45
46
47
48
49
9750
5i
52
53
54
55
56
57
58
59
12 3 4
5 6 7 8 9
8i65|82fo
7717
86i3
9060
8657
9105
9508 9553 9597 9643 9687
• 9955
9870403
o85o
1298
1745
2192
2640
3087
3534
3981
4428
4875
5322
6216
€663
7100
7556
8co3
84508494 85
385o
4396
4743
5i88
8896
9343
• 9789
9680236
o68a
1128
1575
2021
2467
2913
336o
38o6
4a52
4698
5x44
5590
60&
648i
7373
7818
8264
8710
9155
• 9601
9890046009
0492 o5j<
0087
i38a
i8a8
9273
2718
3i63
36o8
4o53
♦000
o448
0895
1 34a
1790
2237
2684
3i3i
3570
4020
4473
4930
5367
58i4
6261
6707
7154
7601
8048
56341567
6080612
6526|657(
6972
7417
7863
8309*8
8754'
9200
9645
o536
0981
i4a7
1872
23l7
2762
3208
3653
4098
78071785217896
8255 8299*8344
8702 8747|87<
91509195
00450090
04920537
09400985
1 387 1432
i834
45i7
4964
5411
5858
63o5
8q4i
9^87
98349878
0280
0727
1 173
'^
2060
25l2
2958
34o4|3449
3895
4341
4787
5233
2326 237
3774
3221
3668
4ii5
67976841
7243 7388
76907735
81378182
8583 86a8
90309075
9521
9968
70o4i4
0816 0861
7908
353
8799
9344
969097349779
7053
8398
8843
93
0180
0625
1071
i5i6
1961
01 34
o582
1029
i477
1924
3538
3984
443o
4876
5332
5768
6214
6660
7To5
75o6|755i
7997
8443
8888
899333
0324
0670
iii5
i56o
2006
345 1
2896
3341
3786
423 1
7041
8389
8837
7986
8434
8881
9284 9329
8o3i
8478
8926
9^74
9732 9776 9821
0179
0627
1074
l52I
1969
2416
2863
33io
3757
4ao5
4652
50995143
55455500
5992 60J7
64396484
6686
7333
7780
8226
8673
0224^0269
0671
1119
i566
201 3
4349*4
4696474
9566
♦012
o45o
0905
1 352
1798
3244
3690
3i36
3583
4029
4475
4931
5367
58t3
6358
6704
7i5o
7596
804 1
8487
8q32
9378
9823
0269
0714
1160
i6o5
3o5o
3495
2940
3386
383 1
4376
6o3i
7S77
7834
8371
8717
9164
961 1
oo57
o5o3
09500994
1396
1843
2289
2735
3i8i
3627
4073
4519
4965
54ii
5857
63o3
6749
7194
7640
8086
853i
8977
9433
9868
o3i3
0759
I204
1649
2095
2 540
2985
343o
3875
4320
80768120
8523
8971
9418
9866
Diir.
I
5i88
5635
6082
4786
5233
568o
6126
6529 6573
6975
7422
7860
83i6
8762
9209
9655
0102
o548
i44i
1887
2333
2780
3326
3672
4ii8
4564
5oio
5456
5902
6348
6703
723
768
7020
7467
7914
836o
8807
9253
9700
0146
o5o3
10J9
i485
1932
2378
2824
3270
3716
4162
4609
5o54
55oo
483o
5277
5734
6171
6618
7065
75ii
7958
84o5
885i
9398
9745
OIOI
06J7
1084
f53o
1976
2423
2869
33f5
3761
4207
4653
5545
01234 56789
DilTJtMalt.
I
2
3
4
5
6
7
8
9
mm
45
.??
180
235
270
3i5
36o
4o5
I
44
2
88
3
l32
4
176
5
220
6
264
7
3o8
8
352
9
396
INT.«sH«h.
Jl»
I
150
LOGARITHMS OF NUMBERS,
Num.
9760
6i
63
63
64
65
66
67
68
69
9770
71
7a
73
74
75
76
77
78
79
9780
81
8a
83
84
85
86
87
88
89
9790
9'
9?
93
94
99
9800
01
oa
o3
04
o5
06
07
08
09
9810
If
la
x3
14
i5
x6
17
18
19
Nam.
12 3 4
5 6
989 4498:4543,4587 463a 4676
4943 49S8 5o32,5o77 5i2i
5388|5433 5477;55ai 5566
5833
6378
6722 6767
5877159221596616011
632 3,6367 64i I
7167
761a
8o57
85oi
8o46
9590
7a I a
7656
8101
8546
6811
7a56
68561
17301
7701 17745
8145 81QO
8590
8990' 903 3
9455J9479
8634
6456
6900
7545
7790
8a34
8679
91 a3
9568
• 983519870 9923|9968;*oia
" iJ,<>3f
072j;o768o8ia
990 oa7o|o3a
?::
II68
1612
ao56
35oo
3944
3389
3333
4^77
4721
laia
i65d
2101
2545
2989
3433
)68o4i2
o857
1257
i3oi
1701 1745
2145)2189
2589I2634
3o3j|3o78
3477i35aa 3566
39ai|3966|4oio
4Jo5;44io|4454
0457
0901
1 345
1790
ja34
3678
3ia3
3877:
4321
4765!48oo|4854|4898
5i64 5ao9l5a5J
56o8
6o52
5653|i
6o96(
5697
6i4i
6496 6540 6585
' ■ ' ' ■ .7028
747a
6940
7383
7837
8371
8714
91 58
9601
99T 0044 0089
6984
74a8
7871
83i5
8758
9203
9^45
o4d8
0931
1 374
1818
336 1
o53i
0975
i4i9
1862
23o5
7916
8359
88o3
9246
9690
5298
5741
6i85
534a
5786
623o
6629 6673
707J
75i6
7960
84o4
8847
9201
9734
3704; 3748
3i47'3igi
3590; 3634
4o33|4o77
4476,4520
4919496315007
5362:54065150
58o5^5849
624716292
oi33|Oi77
057610621
1020 1064
i463 i5o7
66oo;6734
7133,7177
7575 7620
80188069
846il85o5
1951
2394
2837
3280
3723
4166
4609
5o5a
5495
58931 5937
6336 638o
1906
2349
2793
3a36
3679
4(32
4565
67796833
7221 7266
7664i77<>H
7117
756i
8004
8448
8891
9335
9778
0333
o665
1108
1 553
1995
3438
3881
3334
3767
4310
4653
5096
5539
5983
6434
6867
73ro
7752
8107
8549859
8i5i;8i95
8oo3 8947 8991
9345 95901 9 <31
• 9788983219876
992 033o'o275jo3i9
0673107 1 7I076 1
8638
9080
9523
9965
o363|o4o7
u8o5io85o
9o3fi
9478
9921
47a*
5i66
56io
6o55
65oo
6045
7300
7834
8270
8733
9168
9613
oo57
o5oi
0946
1390
i834
3378
3733
3167
36ii
4o55
7 89
4765 4810 4854|4899
4942
5aio 5355 5399 5344
5655 5699574415788
6100 6144 6189 6333
65456589663446678
7434
6989I7034 7078
7478 7533 7567
7879 7923 7968
8333 83688413
8768
931a
9657
OIOI
o546
0990
1434
1878
3323
3
3311
71 a3
76738
3655
4099
44994543
4987
88138657
9357 9301
9701 9746
01460190
o5ooo6j4
10J4 1079
1479 1 533
1923,1967
3367I3411
113856
3355.3300
36993744
41434188
4587'4632
5o3r5o76
5386j543i|5475 553o
583o 5875'59f9 5963
6374
6718
7161
7605
8049
8493
8o36
9823
0366
63i8
676a
7306
8093
8537
8980
6363 6407
68o6,685i
7350,7395
764976947738
8137
858i
9035
8182
8635
9069
80f3
8457
8901
9346
03J5
0679
1134
i568
30I3
3456
3900
3344
3788
4a33
4676
5l30
5564
9424|9468.95i2
9867 9911
o3ioo355
0700:075410798
ii5j 119711241
1596I16401685
303930833138
3483 3537 3571
3925 3970301 4
336934133457
38 1 2*3856 3900
4355:43994343
9956 #000
03990443
4697;4742'4786
5i4o,5i85 5229
5583,5627!5679
6026,60706115
64696513,6557
0842
1386
1739
2172
a6i5
3o58
35oi
3944
4387
0887
r33o
1773
3ai6
a66o
3io3
3546
3989
443a
483o4875
537315317
5716
6911
7354
J
69567000
. 7393:7443
7797,784117885
8239,828483281
8682187368770
61596203
9"4 9169.92' 3
9567 9611 9655
♦009 oo53 0098
o45i|o496|o54o
09940933,0932
6603
7044
7487
7929
83-' 2
881 5
5760
6646
7088
753 1
7974
8416
S859
935719301
96999744
0142
o584
1036
0186
0638
1071
12 3 4
5 6 7 8 9
Di.r.
Dur.
I
9
3
4
5
6
7
8
9
45
iS
180
335
370
3i5
36o
4o5
I
a
3
4
5
6
7
8
9
44
88
i3a
176
a64
3o8
353
396
DifT.JkHalL
LOGARITHMS OF H U M B B R S .
151
9820
31
33
33
35
36
37
36
98 3o
3i
33
33
34
35
36
37
38
39
'42
43
44
45
46
47
48
49
9B5o
5i
53
53
54
55
56
57
58
9860
61
63
63
64
65
66
67
68
69
9870
71
73
73
74
75
76
77
78
79
1 2 3 4
iif5 ii59i3o3ji348
5 6 7 8 9
1 557
»999l
3441
3884
993 OJ
21 57
3480
3931
4363
48o3
5344
5685
6136
6556
7007
7448
7888
8339|8
1601
30443088
3486
3938
38i3
4354
4696
5i38
i646
353o
3973
33703414
3856
4298
474o|4784
5 183
5579'5634
6o3i
6463
6905
7346
7788
6o65
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3378
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1981
3433
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3260 33o4
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4142 4i86
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Dur.
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Dirr.
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DifllftlfQlt.
152
LOGARITHMS OF NUlCBEas.
Num.
•680
8i
8a
83
84
85
86
87
88
89
9890
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92
$
97
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22
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26
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32
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Nam.
1
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8(X)9 8o53
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5 6 7 8 9
Dim
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1 524
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240212446
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x547
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3855
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90449088
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1284 56789
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INC
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TABLE
QW
LOGARITHMIC
SINES, TANGENTS, &c.
158 LosAaTrHHio •inis, taksinti, itc
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AN TILOQARITHMS.
fCu
33ii3
33189
33266
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.521
.523
.523:33343
.524133420
1
.52533497
.526 33574
.527 3365 1
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.531I33963
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.534'34t98
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345 14
34594
34674
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.546 35 156
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.54335318
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■.55r>,35487
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.552'35645
.553 35727
.55U5810
.555 3~5892
.556 35975
.557'36o58
.558j36i4i
.55936224
.56o;363o8
.56 1 136392
.562i 36475
.563136559
^564:36644
.565,36728
.566, 368 1 3
.56736898
.568 36983
.569I 37068
.570 37154
.571 37239
.572 37315
.573 3741 I
.574 37497
.575
.576
.577
.578
579
L.
3-584
37670
37757
37844
3793 r
!fiiiii
Proportional Puta.
1
8!
8
8
7
7
7
7
8
7
8
8
-f
I
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
9
8
8
I
8
8
8
8
9
8
9.
t
9
8
8
5i
9,
9
9
9
9
5!23
6' 23
23
23
23
23
23
23
23
6124
6' 24
5I23
5,23
624
6! 24
623
624
624
624
6; 24
6-24
6' 24
624
624
624
624
24
24
25
H
25
25
25
25
6:24
7 25
25
25
25
25
25
25
26
26
25
26
25
25
^25
7! 26
7I25
26
26
26
26
7J26
826
826
8 26
827
2' 3
4 6 6 7 I 8, 9
38
39
38
38
38
38
38
39
3i
3i
3i
3o
3o
3^
3i
3i
3i
32
3i
3i
32
ii
37
3i
32
32
32
32
32
32
32
32
33
32
33:41
33 4i
32
33
46
46
46
46
46
46
46
47
3946
39J47
3947
3946
39I47
4o47
??|47
3947
3947
4o48
4o:48
4o'48
4o'48
4o>48
4o48
4o;48
4ol48
4i'49
4i49
49
49
49
4|
4i|49
3314 1 49
334i'49
33:41 |5o
33 4i'49
3l'4i^5o
33'4r5o
5o
5o
5o
33'4i
33;42
34:42
33 4a'|5o
33i4ii5o
3442 5i
34 43 5i
34
42'5o
34 43 51
34
34
34
42
42
42
5i
5i
5i
34'43 5i
3i4-> 57
34 43 52
43,52
43 52
43:52
43 52
44 52
44'52
4453
44:53
34
35
35
34
35
35
35
35
54|6i|69
54J6269
54'6i69
53 6ii69
6i^|69
53
54
54
61
62
54!62
54162
5563
55163 71
54'62
55:63
69
70
70
71
70
70
55,63 71
55,63:71
55:6371
55 63 71
56
56
56
71
72
7a
7a
7a
7a
63
64
6^*72
56!64
56|64
56164
56164
5664172
5765,73
576573
65,73
65,74
73
74
74
74
74
57
57
57
58
65
(36
57^66
58'66
58'66
57'66]74
5"8'(>)J75
586675
58
58
661 75
67
5967
"58 67
58 67
5968
5968
75
75
75
75
76
76
5967
5968
5968
5968
59 6S
60 68
5
76
76
76
76
7
77
59r)8>7
6069,77
60 69' 77
606978
61 69 78
6(»69 78
61
61
6i'70
62 70
y' ft"
70,79
7079
79
79
L.
.58o
.58 r
.582
.583
.584
ProporUoMiI Parte.
.585
.586
.587
.588
^589
.590
.591
.592
.593
^4
.595
38019
38107
38io4
38283
38371
38459
38548
38637
38726
388i5
38905
38904
39084
39174
39264
39355
.5^139446
.597I39537
.598I39628
.5991 39719
39811
.600
.601
.602
.6ci3
.604
39902
39994
40087
4or79
.6o5'4o272
.6o6|4o365
.6o7|4o458
.6o8 4o55i
^^609
.610
.611
.612
4o644
40738
4o832
40926
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.6i5 41210
.6i6 4i3o5
.617
.618
.619
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41495
41591
.62041687
.621141783
.622 41879
.62341976
.624 42073
.625 42170
.62642267
.627 42364
.628 42462
.629 42560
.63«) 42658
.63 i
.632
4275fi
42855
.63342954
.63443o53
.635;43i52
.636'4325i
.637'4335i
.638 43451
.639 43551
L.
!f«m
Propoftioual Parte.
1 2 3
\
9
9
17
17
18
18
17
[36
a6
27
27
26
18127
9{i8l26
17126
27
9|i8
8ii"7
918
9
9
10
27
26
27
i8'27
18J27
9'i8 27
9fi8i27
911827
918
_9'i8
"918
10! 19
I O' 19*28
9'i8l27
9 19=28
9118.28
9I1827
28
28
28
27
28
27
28
918
9
10
'9
!9
.,'9i28
9I19I28
9|i9i28
10|I9!2Q
~2 19128
28
29
9 '9
019
o; 201 29
o|i9'29
1929
1929
20;29
oi 19.29
91929
'9^9
1929
20[3o
20 29
o> 1 9 29
o 20 29
0,20:30
20*29
1929
29
3<)
020
o'2o)3o
o 20 3o
0J20 3o
o2o!3o
FT'T
35
35
36
36
35
35
35
35
36
h\ 6
4453
43 52
4453
45 53
4453
3644
53
4453
44 53
44 53
4554
354453
36 45 54
36,45 54
36145 54
37146 55
36*45 54
36 45 54
36 45 5-^
36 45 55
3j;46 55
36'46 55
37 46 56
37 47 56
37 46 55
37i^56
37I46 55
37 46 55
37 46 56
37
38
38
38
38
47 56
47,57
47|56
4756
38-47 57
38 48 57
47:57
47'57
47i57
48.57
48.58
38
38
38
39
38,48,58
38i48;58
39 48 58
394958
39 48158
38!48,58
6f
61
70
9
79
39148 58
39|49!58
39,49.59
39^49; 59,
ll ^??
39 49|59
40 5o 59
3949
3949
3949159
4o5o
4o5o
4o5o
4o5o
4o5<
^9
60
60
60
60
60
6a'7i
6a
62
7079
6a
62
6a
62
71
80
80
7079
71
71
80
80
7180
17180
63 72I81
62
63
7«
7a
63'72
80
81
81
63-72'8i
64
63
63j7a
63!73
64173
64173
73|8a
73 8a
8a
8a
8a
83
64|73,8a
65'7483
6517483
6474183
65;7^i83
657'4!83
65-7483
65.7484
65!75'84
66:75 85
66'75'85
66 75'85
66;76.85
6776.86
66!76,85
66i?6 85
67'76.86
86
86
6777
67 7_7 _
6777^86
67177 87
68.78'87
687787
68 77187
68
68
7787
78ifl»
69!7888
667888
68*78|88
69;79,*8
69179,89
6979,89
70
70
7?j8?
79,89
8090
708090
To'8090
7o|8o9ii_
7' 8' 9
Proportioaal Autt.
ANTIL06ARITBM8.
297
.640
21am
Froportional PartJi.
.700
Nam
Proportioaal Parts.
1 2
I0t20
3
3o
4 5 6
4o 5o6o
7 8
7080
9
90
1
11
2,3
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4173
418)
4197
014309
J 4331
4j34
4a46
4358
4370
4383
4394
4307
COM.
9.8
o64i3
06401
o
I
I
6389
6377
6366
6354
6343
633o
63i8
63o6
6395
6383
6271
3,6259
36247
1,6^
316223
3 6211
46200
6188
46176
4,6164
46i5a
516 1 4o
56128
5
6
6116
6104
6092
6080
66068
6(^
6
6
7
7
in.M
m.i
tan. cofiec
8
8
8
9
9
9
9
9
o
o
o
o
o
I
I
I
I
I
a
a
3
6o44
6o3a
6oao
6008
5^
5964
597a
5960
5948
5936
5934
591a
5900
5888
5^876
5864
585 1
5839
58a7
58i5
58o3
5791
5779
5^
5754
574a
5730
5718
5706
5693
440
a.
9.8
■in.
_9.8_
4177
4190
4ao3
4ai6
4339
4a42
4355
4369
4383
4295
43ott
4331
4334
4347
43()<)
4373
4385
4398
44m
44M
4437
445o
4463
4476
4489
45o2
45i5
4^28
4540
4553
4566
4579
4592
46o5
4618
4^)37)
4643
4656
4669
4682
4694
4707
4720
4733
4745
d.
cosec
10. 1
5823
o58io
05797
5784
5771
5758
5745
5731
5718
5705
3 5692
315679
35666
3] 5653
3 5640
8484
85o9
8534
856o
8585
8610
8635
8661
8686
8711
3,5637
315615
45602
4 5589
415576
45563
555o
5537
5534
5|55n^
5 549«
5485
547a
546()
5447
8737
8763
8787
881a
8838
8863
8888
8913
8939
8964 '_8
898o:'8
9015 9
9040! 9
9065
9090
6
6
6
6
6
7
7
7
7
9116
9141
9166
9191;
9317
5434
543 T
5408
5395
5383
8 53^0
85357
815344
85331
85318
«in.
460
4758
4771
4784
479^
4809
4822
4835
4847
4860
4873
4885
4898
4911
4923
4936
4949
9.8
CO*.
1340^
9
9
9
9
_?
f)
o
o
o
I
d.
53()()
5293
5280
5267
5255
524?
5ai
5ao4
5191
5778
5i65
5i53
5i4o
5ia7
5ii5
5l02
5089
5o77
5o64
5o5i
7oj'
PCC.
tan.
9.9
O
O
I
1
2
2
3|
3
4
4
5
5
5
6
"6
7
7
9343,
9367,
9393
9318
9343'
9368
9394'
94 '9
9444!
9469.
9495
9520
9545
9570
9596
9621
5229 9646
6967V
o
I
I
I
a
2
3
3
3
4
4
o
o
c
c
('
c
c
c
o
(
5o
5o
6
6
6
7>'
8
8
/
81
I
o<
o-
00 .*-
.V
V :
t
o43v>
o4o4
20
20
9697
9722 21
9747J31
9773,31
9798 22
9823 22
9848,23
9874/^3
989924
9924 a4
994924
997525
ten
9.9
0379
o354
0328
o3o3
0378
0353
0237
oaoa
0177
Ol53
0136
0101
0076
oo5i
ooa5
a5oooo
cot. I d.
lao
tan.
4838
485o
4863
4875
4888
4900
4913
4936
4938
495!
4963
4976
4#
5ooi
5or4
5o36
5o39
5o5i
5ioo 15
10.1
coiecl d.
■m.
450
\ t
'AVt^A''' averse table,
Aw" *
OR
TABLE
or
i9
^4
7
/
^ * • f-»»
INGE OF LATITUDE
AITD
DEPARTURE.
324
TRAVERSE TABLE.
Cotirse
pOlllM
o
i
i
I
«1
1 5'
So'
45'
1 5'
3o'
45'
1 5'
3o'
45'
1 5'
3(y
45'
1 5'
3<>'
45'
1 5'
3o'
45'
1 5'
45'
70
1 5'
3o'
45'
1 5'
3&
45'
1 5'
45'
1 5'
3c)'
45'
IP
1 5'
3o'
45'
1-20
ly
3o'
45'
1 5'
3o'
45'
14«
1 5'
3o'
45'
150
poiot<l dffg.
Course
Dist. 1
dif. lat. I dep.
I .(XXKM) 0.00000
0.99999
9V990|
99991 1
0.999^15
99976
99966
9'>953|
0.99930
999"
99005
99885
0.99880
0.99863
998301
998 1 1
99786
0.99756
997i5
99697
99**57
0.99619
99680
99540
0.99518
9g4o-?
0.99452
994<»6
99^57
99307
0.99^55
99^00
99'4^
99f>87
o.9y«)27
989651
0.9S918
98907
98836
o 98769
98700
9S679
9855^)
<>.9Si8ii
98404
98J35
98245
0.98163
0.98079
9799^1
9-'9o5|
0.97815
97793
97630
97534
0.97437
97338
97337
97134
(^.97030
0.97003
96933
96816
96705
0.96693
[ 00436
00873
oi3oo
0.01745
03 1
036
181
»i8
o3o54
0.0349
03926
04063
04798
0.04907
0.06334
06669
061 o5
06640
0.06976
0741 1
0784::
o8i8i
0.08716
09160
09686
0.09802
10019
0.10453
urfJ87
Il330
II754
0.I2I87
13630
i3o53
1 3485
o.i3qi7
14I40
0.1 467 J
1 478 1
16312
0.1 5643
i6c)74
i65.»6
i6o36
0.17J65
17794
18224
18662
0.19081
0.19609
19037
3o364
0.20791
31218
a 1 644
32070
0.23496
33930
23345
33769
0.24192
0.24298
2461 5
36o38
36I601
0.36882
d«p.
dif. Int.
Dint. 1
Ditt 2
0.00<)<K»
00873
01745
03618
0.03490
04363
o5235
06108
0.06980
07862
08724
09696
0.09814
0467
1339
o.
di£ lat. I dep
3.00000
•99998
99992
99983
.99970
99962
99931
999^7
.99878
99846
99810
99770
.99760
.99726
99678
99627
99672
.99613
99460
99383
993 1 3
.99239
99161
99079
.99037
98994
.98904
9881 T
98714
98614
.98609
98401
98289
98173
.98064
3.)
979
97835
97803
97672
.97538
97309
97267
97111
.96962
96808
96661
96490
.96326
.96167
96809
.96630
96446
96269
96068
.94874
94676
94474
94268
.94069
.94006
93846
9363o
93400
1.93186
J951
(823
3o8i
3
4
6692
6662
743 1
83oo
'^
2oo38
0.20906
21773
32641
23607
0.24374
26240
26106
26970
0.27635
28699
0.29346
39662
3o425
0.3 1 287
32149
33oio
33870
o.3473o
36589
36447
373o5
0.38162
0.39018
39874
40728
0.41682
42436
43288
44139
0.44990
46840
46689
47537
0.48384
o.485o6
49231
60076
60920
0.61764
dep. I dif. lat.
D.^.t. 2
Dist. 3
diC lat. dep.
3.00000
2
9
99974
a.99964
99029
998?7
99860
2.99817
99769
99714
99666
•» .99639
2.99689
99616
99440
99358
2.99269
99175
99075
98970
2.98868
98741
98619
3.98666
98491
3.98357
98217
98072
97921
a .97764
97601
97433
97260
2.97080
968
3.967
96706
96608
3.96307
96886
96667
a. 9644 2
96312
94976
94736
a.94488
2.94236
93977
93714
2.93444
93169
9288
9260
3.93311
9aoi4
91711
9i4o3
a.91089
3.91009
90769
90444
90114
3.89778
o
I
o.oooot*
01 309
02618
03927
0.062 36
06644
07863
09162
0.10^70
1 1 778
i3o86
14393
0.1472
0.16701
17008
i83f5l
19621
0.20927
22233
23538
24842
0.261 4"
2746<^
28764
0.39406
3«x>66
0.3 1369
32661
33961
36-»6i
0.36661
37860
39168
4o456
0.4176?
43o48
0.4401
4434
45637
0.46930
48223
4951
5o8o6
0.6209'f
53383
54671
56967
0.67243
0.68627
6981
61093
0.62374
63663
64932
66209
0.67486
68760
70034
7i3o6
0.73677
0.72894
73846
75ii4
76381
0.77646
?
o
dnp. j dif. laU
Dim. 3
Dif t. 4
dif. lat.
4.CKXX)0
3.99996
99985
99966
3.99930
99906
99663
99813
3.99766
99692
99619
99530
3.99618
3.99462
99367
99264
99»44
3.99026
98900
98767
98626
3.98478
98322
98168
3.98074
97987
3.978<»o
9762S
97439
97337
3.97018
96803
96678
96346
3.96107
96861
3.96671
96606
95345
3.95076
9^799
945 1 4
94233
3.93923
93616
933o3
92960
3.92661
3.92314
91970
91618
3.91269
90892
90618
001 37
3.89748
89352
88948
88537
3.88118
3.8801 3
87692
87360
86816
3.86370
dep.
di>p.
0.00000
01746
03491
o5336
0.06981
08726
io47»
I22l5
0.13960
i57»>4
17448
1919
0.19627
0.20934
32677
34419
a6i6i
0.27903
39643
3i384
33133
0.34862
366oi
383381
0.39207
40076
0.4181 1
43547
46281
47015
0.48748
5o48o
5a3io
53940
0.66669
67397
0.66692
59134
60849
0.62674
64397
66019
67740
0.69469
71177
73894
74610
0.76334
0.78036
79747
81457
o.83i65
84871
86676
88379
0.89980
91680
93378
96074
0.96769
0.97193
0.98461
1.00162
oi84i
1.03538
dif. Uu
Di9t. 4
Course
def
45'
30^
i5'
45'
3o'
1 5'
88^
45'
30*
i5'
87®
45'
So'
i5'
86^
45'
3o'
1 5'
45'
3o'
15'
84^
45'
30^
i5'
83°
45^
30*
ly
88°
45*
3o'
i5'
81°
45*
3o'
15'
8O0
45'
3o'
1 5'
79°
45'
3o'
1 5'
78°
45'
3o'
1 5'
77°
45'
3o'
1 5'
76°
45'
So'
15*
75°
pmnu
T
7l
74
7i
6i
def.
pmnu
Course
?1
p««l.
d»t.
dir UL
i'V-
.^J^
diCi... d.p.-Uli'.l... t-f.
ditTsi:
^dET
"d;r"p"i"u|
0°"
5.0000
e.oooolo.oiioii
7.0000 0.0000 8.0000 0.0000
i;^
W° 1 B 1
i5'
S-oooo
0118
59999
0363
5.9999 o3o5
"^\°^
"■^
45
3o'
4-9W8
o436
9998
o534
9997 061 (
9997
07B5
3(1
45
1°
we
o6ii
0^873
9995
5.999 r
D785
0.1047
^^oZ'.
9993 "o47
>^
.178
0.1571
i5'
390
i5'
1091
9986
i3o9
Si! ,s=,
'■9981 "nt
99^9
:f5^
iK
30-
W83
.3^9
9979
.5^
S,»' ,»3.
9973 109,
^
3o'
45'
9977
isl?
i83i
9,6,. .,3a
9963 3443
3748
1 5'
3°
49970
0.1745
5W63
°-'X
S.99S. 0.443
7.9951 0.3795
B.9945
0,3. 4 1
**'
i5'
996"
1963
9954
994S >7«
9938| 3,41
993.
3533
45'
30-
.^'
31S1
o.J^?l
9943
993.
36,7
1879
9933, 3.>5J
p li^.
,io^
s;s
3o'
15-
i
5.99j8!o.394i
5,99„io3435
;*,o:X^
8.989'
0,4416
7l
3»
4:9931
0.16.7
j.99180.3140
5,9904 o,3,",<id
8.9877
37°
1 5'
1835
^S
34o3
98711 4535
98^5
5l03
45'
}o>
9^
3o55
3663
9»9
4373
985 1; 4884
9833
'd^
3o'
4S'
?B73
o^?8S
9«5,.
4578
7.9805 o,558i
9807
i5'
4=
4.987B 0.3488
59854
6.98.,
o.48«3
B.9781
0.617B
*o
1 5'
^863] 3705
9835
4447
9».«
5.88
97801 5939
9753
6670
45'
So-
?846 39, J
98.5
4708
9784
549a
9753 6377
9733
706.
3o'
4S'
98JB 4i4o
9794
4968
97C0
5797
9715I 663S
9691
7453
i5'
5=
i5'
4 $810,0.4358
9790 4575
5.9771
9748
0,5^
6-9731
0.6101
e4o5
'■^i^t?^
B.96i8
9633
0.7841
8335
45-
ao-
9770 479'
9734
?W
6709
9633] 7668
95W6
8636
3o'
1
'■&r^
5.97 J 1
0.588.
6.96630.6861
7.96.5I0.784.
8,9567
0JI813
7i
45'
969S
60.,
?64»| 70,3
9597I 8oi5
9547
9017
1 5'
4.9756,0.5316
5.967T
0.6373
6.96,70.73,7
7.956J o,a36i
8.950-
0940B
M"
i5'
5443
9643
6533
9584
763.
95i5j 8709
94H6< 9u56
94fij
:m
45
Itf
^1^
5660
96.4
6793
9550
79=4
9-1-^1
3o'
45'
?65i
5H77
9584
to'!.
,5,5
833H
9445I 9io3
9376
0578
.5
70
4.5fi'7
0.6093
5.$553
0.7313
69478
0.853.
7.9404
0.9750
8.5339
■s
SJ"
i5'
63io
95ao
7571
9440
8834
9360
T.0096
9180
45
Stf
^
9487
7633
940.
91 37
9316
0443
9i3j
1747
3o'
9543
4.9513
9483
0^
9451
5.93.6
9379
B6ia
9361
9440
0.9743
T.0044
9169
7,933.
9.73
0788
1. 1134
,.;S
9' 78
B.9114
9oe^
3i37
1.1536
3914
i5'
1
4.9459
0.7337
5935;
0-8804
S.934I
7.9-34
..33o5
71
So-
545,
7390
934.
B869
933.
o347
913L
1835
895:
33o3
3o'
45
i5
?4ra
4.9384
o35o
7656
-0.7833
8037
5.9J61
91 JO
■IS
9645
9.85
S.9I38
1.0950
laSl
9069
'1^
■ii
369.
.,4079
4467
i5'
SI"
3o'
45'
9J78
8>5>
8467
iw,
0.99^3
^
1553
i854
K
13
8766
8700
4854
5341
3o'
i5'
10°
i5'
4.9:140
0.8681
8897
5.90B8
SI
'■^'
'^l
1.3.55
1456
7.8785
8733
•X
B.B633
8564
1.5638
601 5
80°
45'
30'
l^ei
9113
0934
SB38
3756
B660
4579
8493
64oi
3o'
45'
9tj3
9336
5»
1191
8773
3o57
8596
49"
8431
6787
1 5'
11°
4.L)i
0-9540
1.1449
S.87I4
..3357
7,853o
1.5365
B.8346
..7173
790
i5'
'&
0.^55
5.8B47
..1705
S.Bf.55
1.3656
7.8461
I.5607
8.817 <
..755«
45'
Stf
0.9968
8795
1961
85,5
395fi
8394
5949
8193
IS
3o'
45'
803
T,6T83
8743
,.«;i
8533
4355
8334
,.^?3
8ii4
i5'
18^
8Si5
[.0396
5.8689
5.8470
..4554
7.8153
B.Bo33
1.8711
79°
i5'
30'
0613
8634
B578
;j2i
B4o6
834.
4853
5r5i
8178
8.04
6974
^;
»
45'
3o'
45'
8767
io35
85i.
3a
8374
5449
8037
?656
7781
1.9863
t5'
13"
4.8719
I.I34B
5.846J
■if.
5.8ao6
..5747
'■^e
•lt&
8,7693
1.0146
770
i5'
866^
.460
84o3
8i37
6044
7604
0638
45'
3o
8618
1673
8.341
4007
6066
6341
7790
8676
75.3
30'
45
8567
1884
8381
436.
7994
6638
9^;^
7431
Iv
i5'
H°
faSrS
5-83.8
.,45i5
6.793 J
..6935
7.7634
T.9354
B.7317
3,1773
76°
1^
1-850)
■i
5.8303
..4579
5.7911'
..700^
7.7603
..9438
8.73o3
^
t5
846i
81 54
£S
7846
733;
7538
..9693
713.
45'
So-
B4o7
.S,J
n
77-70
7517
7453|3.O03.1
7133
3o'
4y
835j
.,3;
5'.76
7693
7833
7364: o36f
7034
19. ,
1 5'
k:^
15a
-d^r
4.8396 r.394'
5,7956:7.553^
6,7615
T.8iT7
diTin:
7.737411.0706
d^, ldlf.U,L
B.6933
S
7r,°
gs
1 tt«r«
Oi.t4
Wfi 1 Di
bai. 8 1 Disi. 9
Col
326
TRAVERSE TABLE.
Course
|>ointit
iU-
ih
li
3i
points
15«
3(>'
45'
1 5'
3()'
45'
1 5'
45'
18'-^
1 5'
3(y
45'
1 5'
45'
i5'
3o'
45'
1 5'
3o'
45'
1 5'
3o'
45'
23^
1 5'
3o'
45'
1 5'
3o'
45'
25-
i5'
3o'
45^
26^
1 5'
3(>'
45'
27^
1 5'
3o'
45'
2d^
1 5'
3o'
45'
1 5'
3o'
45'
30<3
Dist. 1
dif. lat.
O
9^)479
9^94<>
0.961 jCyo
9600 •'»
9588i
95757
0.95^0^0
o.95r)3()!o
9550^
95373
95 i 4(1
0.95 f 060
9497* »|
9483.»,
946931
0.9455^ o.
94^091
943fHj
0.941540.
94»i8l
o.939(T9'o
93819
93r/i7
93514
0.93358 o
93201
930 'ia
99881
0.97718 o
9'5S4j
0.923880
92220!
0.920500
91879
91706
9i53i
0.913550
91 176
9099*)
90814
0.9063 1 jo
90446I
o.9o3(^ (
90259
9<X)70
0.898 ""9
8(/vH-7
89'(93
89298
o 89101
88902
88701
88-^99
0.882950
0.88192 o
88089
87H82
87673
0.87462 'o
87250
87o36
86820
o.866o3,o
o
o
dp|{.
Course
dep.
25882
263o3
26724
27144
275^i4
27983
28402
28820
29f)28
29237
29654
3oo7i
3o486
3<.)902
3i3i6
3i73(»
32144
32557
32969
33381
33689
33792
34202
34612
35o2!
35^29
35837
36244
3665o
3-o56
37461
37865
38268
3867!
39073
394^4
39875
4'»275
40674
41072
41469
4i8^i6
42^62
42657
42-'56
43o5i
43415
43837
44229
41<>20
45oio
45399
45^87
46175
46561
.4694-
47 1 4o
47332
47716
48099
.48481
48862
49^42
49^)22
.5()OfX)
Diat. 2
dif. lat.
dop. I dif. Int.
Dist. 1
.93185
92957
92726
92491
.92252
92010
91764
9i5i4
dep.
0.51764
52606
53448
54288
o.55i27
55966
568o3
57639
.91388 o.58o57
91261I0.58474
9icK)4
90743
9^^479
.0021 1
09940
896(^5
59308
6oi4i
60073
0.61803
62633
6346 1
89386' 64288
.89fo4!o.65i 14
88818 65938
88528 66761
.88309 o.6-'378
88235| 67583
.879390.68404
87638 69223
87334 7004 1
87027 7o858
867160.71674
864o2 72488
86<i84
85762
733oo
74111
.85437 0.74921
85io8 75730
.847-76 G.76537
84440 77342
.84101 0.78146
83758 78949
83412 79750
83o62 80549
.827090.81347
82352 82144
81992 82939
81629 83732
.812620.84524
80891 853 1 4
.8o798'o.855ii
8o5i7 86102
8oi4o 86889
.79759 0.87674
793-5 88458
89240
90020
789«
78596
.78201
77803
77402
76998
.7659(
0.90798
91575
92350
93123
0.93894
.7638.1 0.942-79
76178
75763
75345
.74924
74499
74071
73640
9 1664
95432
96198
0.96962
97724
98485
DiMt. 3
dif. lat. I dftp.
2.89778!0.77646
89436
89089
88737
2.88379
8801 5
87646
87271
2.87082
2.86891
865t/)
86i!5
H5719
2.85317
84910
84497
84079
78909
80172
81432
0.82691
83940
852o5
8645o
0.87085
0.87712
88962
90212
91459
0.92706
93949
96101
964J2
2.836 5^")|0.9767o
83227 098907
82792
2.82463
82353
2.81908
81457
81002
8o54i
2.80074
79602
79126
78643
2.78155
77662
2.77164
76660
2.76161
75637
76118
74593
2.74064
73529
729,88
7'443
2.71802
7i337
2.71197
707-6
70209
2.69638
691^) 2
68480
67894
2.67302
66706
66io3
65496
2.64884
2.645-6
64267
63646
63oi8
2.6238<i
61749
61 107
0.99243 60460
2.598<-»8
.7320611.00000
dup. I dif. lat
dep
i.oor42
1.01067
01376
1 .02606 3
o3835
06062
06287
1.07610
08731
09960
11 167
1. 1 2382
13596
1. 1 4806
i6oi3
1.17219
18423
19626
20824
1.22021
23216
24408
26698
1.26786
2797 r
1.2826^
29163
3o334
i.3i6i I
32687
33869
35o3o
1.36197
3-362
38526
39684
1 .4084 1
i.4i4t9
4199b
43 1 48
44297
1.45443
46686
47727
48866
1.60000
Dist.
dif. Int.
DiKiry
Dis
dif. lat.
3.86370
86916
86452
84982
3.845o6
84o2o
83628
8 3029
3.82776
3.82622
82008
8i48t
80968
3.80423
79880
79329
78772
3.78207
77636
77067
3.76618
76470
76877
76277
74669
74o54
73432
72803
72167
71624
3.70874
70216
3.69662
68880
3.68202
67616
66824
66126
3.66418
64706
63986
63267
3.62623
61782
3.61696
6io34
60279
3.69618
58749
57974
67 1 92
3.664o3
66607
54804
53996
3.63i79
3.62-f)9
62356
61627
60691
3.49848!
489981
48i42{
472801
3.464 1 oj
dep. I
Die
d«>p.
.03528I
06212
06895
08676
.10266
1 1932
i36o6
16279
.161 14
•:^?
20282
21946
.23607
26266
26922
28676
.30227
31876
33623
.34756
36167
.36808
38447
4008 3
41716
.43347
44975
46000
48223
.49S43
61469
.63o73
64684
.66292
67898
69600
61099
.62696
64288
66877
67464
.69047
70627
.71022
72204
73778
.75348
76916
78479
80039
.81696
83i5o
84699
86246
.87789
.88669
89328
90864
92396
.93924
95448
96969
.98487
.00000
dif. lat.
Coonte
deg. potiju
75^
45'
So'
1 5'
740
45'
So'
1 5'
46*
3o'
1 5'
72^
45'
3o'
1 5'
710
45'
3o'
ly
70'^
45'
30/
1 5'
45'
30^
1 5'
45'
30*
1 5'
670
45'
30/
1 5'
45'
So/
1 5'
45'
So'
i5'
64"
45'
3o'
i5'
63-*
45'
3o'
15'
62°
45'
3o'
1 5'
61°
45'
3o'
1 5'
60°
6i
6i
5}
51
def.
points
Coura^
TRAVERSE TABLE. 327 |
Course |
Disc 5 1
Dist. 6
Dist. 7
Dist. 8
Dist. 9 1
Course 1
poiiiu
deg.
15^
dif. lat.
4.8296
8239
dep.
diC lat.
5.7956
dep.
dif. lilt. dep.
dif. lat.
7.7274
dep.
dif. Int.
dep.
dog:.
poiniit
1.2941
1.5529
6.7615 I.8117
2.0706
8.6933
2.3294
75°
1 5'
3i52
7887
5782
7535
84l2
7i83
1042
683 1
3673
45'
3o'
8182
3362
7818
6<j34
7454
8707
7090
1 379
6727
4o5i
3o'
45'
8123
3572
7747
6286
7372
9001
6996
1713
6621
443o
i5'
16^
4.8o63
1.3782
5.7676
1.6538
6.7288
1.9295
7.6901
2.2o5l
8.65i4
2.4807
74°
1 5'
8002
3991
7603
6790
72o3
9588
6804
2386
64o4
5iS5
45'
3o'
7941
7879
4201
7529
7041
7117
1. 988 1
6706
2721
6294
556 1
3o'
45'
44io
7454
7292
7o3o 2.0174
6606
3o56
6181
5938
i5'
li
4.7847
1.4514
5.7416
1.7417
6.6986
2.o320
7.6555
2.3223
8.6125
2.6126
6i
17°
4.7815
1. 4619
5.7378
1.7542
6.6941
2.0466
7.65o4
2.3390
8.6067
2,63i3
73°
i5'
775 1
4827
73oi
7792
685i
0758
6402
3723
5952
6689
45'
3o'
7686
5o35
7223
8042
6760
1049
6297
4o56
5835
7064
3o'
4^
7620
5243
7144
8292
6668
i34i
6192
7.6085
4389
5716
7438
1 5'
18^
4.7553
1. 545 1
5.7063
1. 854 1
6.6574
2.i63i
2.4721
8.5595
2.7812
72°
1 5'
7485
5658
6982
6899
8790
6479
1 92 1
5976
5o53
5473
8i85
45'
3o'
7416
5865
9o38
6383
2211
5866
5384
5349
8557
3o'
45'
7347
6072
6816
9286
6285
25oi
5754
57i5
5224
8930
1 5'
ID-*
4.7^76
1.6278
5.673 r
1.9534
6.6186
2.2790
7.5641
2.6045
8.5007
2.9301
71°
1 5'
7204
6485
6645
1. 978 1
6o8()
3078
5527
6375
4968
2.9672
45'
So'
7 1 32
6690
6558
2.0028
5985
3366
54ri
6705
4838
3.0043
3o'
>}
4.7077
1.6844
5.6493
2.02 1 3
6.5908
2.3582
7.5324
2.6951
8.4739 3.o32o
6i
45
7059
4.6985
6896
647'
0275
5882
3654
5294
7033
4706
o4i3
1 5'
20O
1. 7101
5.6382
2.o52I
6.5778
2.3941
7.5175
2.7362
8.4572
3.0782
70°
1 5'
6910
73o6
6291
0767
5673
4228
5o55
7689
4437
ii5i
45'
3o'
6834
75io
6200
1012
5567
45i5
4934
8017
43oo
i5t9
3o'
45'
6757
77i5
6108
1257
5459
4800
48ii
8343
4162
1886
t5'
310
4.6679
1.7918
5.601 5
2.l5o2
6.5351
2.5o86
7.4686
2.8669
8995
8.4022
3.2253
69°
1 5'
6600
8122
5920
5825
1746
524i
5371
456 1
388 1
2619
45'
3o'
6521
8325
\%
5129
5655
4433
9320
3738
2985
Sof
45'
644o
852«
5729
501-7
5939
43o5
9^^45
3593
335o
iV
22^
4.6359
1 .8730
5.5631
2.2476
6.4903
2.6222
7.4175 2.9969
8.34473.3715
68^
i5'
6277
8932
5532
2719
4788
65o5
4043 3.0292
3299 4078
45'
1
3o'
4-6 [94
1. 91 34
5.5433
2.2961
6.4672
2.6788
7.3910
3.061 5
8.31493.4442
3o'
6
45'
6110
o336
5332
32o3
4554
7070
3776
0937
2998 48o4
i5'
23°
4.602 5! 1. 9537
5.523o
2.3444
6.4435
2.7351
7.364n
3.1258
8.28453.5166
67°
1 5'
5940
9737
5l27
3685
43i5
7632
35o3
i58o
2691
5527
45'
3o'
5853
1.9937
5o24
3925
4194
7912
3365
1900
2535
5887
3o'
45'
5766
2.0137
4919
4i65
4072
8192
3225
2220
2378
6247
1 5'
240
4.5677
2.o337
5.48iJi2.44o4
6.3948
2.8472
7.3o84
32539
8.2219
3.66^6
66°
1 5'
5588
o536
4706
4643
3823
8750
2941
2858
2059
6965
45'
3o'
5498
0735
45o8
4489
4882
3697
9029
2797
3175
1897
1733
7322
3o'
45'
5407
0933
5r2o
3570
9306
265 1
3493
7679
1 5'
•25°
4.53i5
2.1 i3i
5.4378
2.5357
6.3442
2.9583
7.25o5
3.3809
8.1568
3.8o36
65°
1 5'
5223
1 328
4267
5594
33i2
9860
2356
4i25
i4oi
8391
45'
H
4.5199
2.1378
5.4239
2.5653
6.3279
2.9929
7.2319
3.4204
8.i359'3.848o
51
3o'
5129
5o35
1526
4i55
583 1
3i8i
3.01 36
2207
4441
1233
8746
3o'
45'
1722
4042
6067
3o49
o4i 1
ao56
4756
io63
9icx>
i5'
26°
4.4940
2. 1919
5.3928
2.63o2
6.2916
3.0686
7.T904
3.5o7o
8.089113.9453
64°
i5'
4844
aii4
38i2
6537
2781
0960
1750
5383
0719*3 9806
45'
3o'
4747
23lO
3696
6772
2645
1234
1595
14^8
5696
o544
4.01 58
3o'
45'
46i^
25o5
3579
7006
2509
i5o7
6<K)8
o368
o5o9
1 5'
27°
4.4550
2.2700
5.3460
2.7239
6.2370
3.1779
7.1281
3.63.9
8.0191
4.0859
63°
1 5'
445 r
2894
3341
747'
223l
2o5i
1121
663o
8.0012
1209
45'
3o'
435i
3087
3221
77o5
2091
2322
0961
6940
7.9831
1 557
3o'
45'
4249
3281
3099
7937
1049
2593
0799
7249
9649
1905
1 5'
28°
4M4i
2.3474
5.2977
2.8168
6.1806 3.2863
7.0636
3.7558
7.9465 4-2252
62°
>i
4.4096
2.3570
5.2915
2853
2.8284
6.1734 3.2998
7.0554
3.77'?
7.9373 4.2426
5i
1 5'
4o45
3666
8399
1662
3i32
o47i
7866
9280
2599
45'
3^
3941
3858
2729
863o
i5i7
3 lor
o3o5
8173
9094
2944
3o'
45'
3836
4049
2604
8859
1 37 1 3()69
7.0I38
8479
8905
3289
1 5'
29^
4.373 r
2.4240
5.2477 2.9f)89]6. 1 223,3.3937
6.997(^ 3.8785
7.87i6|4.3633
61°
1 5'
3625
443 1
235o
9317
1075
4203
9800
9090
8525
3976
45'
3o'
35f8
4621
2221
9545
0925
4470
9628
9394
8332
43i8
3o'
45'
3410
481 1
2092 2.9773
0774
4735
9456
3.9697
8i38
4659
1 5'
^°
4.33oi
dep.
2.5ooo
5.1969
d»j|i.
3.0000
dif. lat.
60622
3.5fKX>
dif. lit.
t. 7
6.9280
4.0000
7-794'
4.5000
dif. lat.
60°
de^.
points
points
dee.
dif. lat.
d«'p.
df>p.
dif. lat.
dep.
-c^.
irao
DiHt. 5 1 Dist. 6 1
Dia
Dist. 8 1
Dist. 9 1
Course |
328
TRAVERSE TABLE.
Coiirie
pointu
^\
3i
3i
31
45'
31°
1 5'
. 3<y
45'
32^
1 5'
3o'
45'
33-
i5'
3o'
45'
34^
1 5'
3o'
45'
•35^
1 5'
3o'
45'
36^
i5'
3&
45'
37^^
1 5'
3o'
45'
3d»
1 5'
30*
45'
1 5'
3(y
45'
1 5'
3o'
45'
41^
1 5'
3o'
45'
iV
3(V
45'
43^
i5'
3(>'
45'
440
1 5'
3o'
45'
450
point*
de».
Course
Di8i. 1
diC lat.
86384
86i63
859^1
0.85773
0.85717
854qi!
85264!
85o35!
o.848o5
84573|
84339I
84i()i
o.8i867
836^9
83389J
0.83147'
0.82904
82659!
8^4 1 31
82i65|
0.81915
81664
8i4ia
81157
0.80902
80644
80 386
o.8o32i
8oia5
0.79S64
796<x)
79335
79069
0.78801
7853a
78261
779«8
0.77715
77439
0.77301
77162
76884
0,76604
76323
76041
7*)7'>^)
0.75471
75 1 84
74896
74606
0.743 1 4
o.74'>95
74022
73728
73432
0.731 35
7a837
72537
72236
0.71934
7 1 63o
7i325|
71019!
0.7071 1 1
dep.
o.5oo(x)
5o377
50754
5i 129
o.5i4io
o.5i5o4
51877
52a5o
5a6ii
0.529*12
5336i
53730
5409
0.54464
548a9
55194
0.55557
0.55919
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22l3l.6
22469 3
22821.5
23226.4
23685.4
242 1 5.4
2484a. I
26600.2
26690.2
27Q95»
3o375.o
infin.
o
I
2
3
4
5
6
7
8
9
10
II
2
[3
4
6
[6
17
t8
1
20
21
32
23
£4
25
26
27
28
29
3o
3i
32
33
34
35
36
37
38
39
40
4i
4a
43
44
45
46
47
48
4?
5o
5i
52
53
54
55
56
57
58
43
cc
333 TABLE FOR COIRBCTING MIDDLE
t»,.,.D..
lIi.
DiffcreoiM of LHiilode.
L.1:
1=
go 30
4= 5= 1 6
- -- TO
10=
1 10 IJO
13=
W 15°
16=
17=
1 0°
i5
_:
;
5
;'i
.3I.;
3i
35
40
45
5i
' 74
i5
t6
5
3o
34
38
43
S
- 7t
16
4
11 Ii5
38
3i
37
43
; SB
4
10 J4
3i
36
41
46
: 56
18
1?
3
16
19 i3
36
3o
34
4o
45
; 53
!?
°
I
J
75
■5
18
"
a5
35
39
33
33
38
37
43
43
i Jo
z iB
i5
17
34
33
36
' 56
jj
.5
17
34
38
33
36
40
<; JS
33
24
°
I
J -
■ 4
■ 4
16
16
'9
73
33
37
3i
37
35
35
g
i i3
34
35
a6
J7
P
I
;
■ 4
■ 4
16
■ 6
19
"
36
36
3o
Jo
34
33
3a
38
<! i3
36
jS
.3
i5
35
39
33
37
/ 5.
38
=9
i3
i5
35
39
33
36
i So
'9
3j
°
]
i3
j3
75
i5
16
18
jl
35
35
18
33
33
J6
36
' 5o
3o
3.
3i
i3
l5
■ 8
35
3B
3i
36
^ bo
3i
33
i3
i5
18
34
37
3i
35
' *9
33
34
i3
i5
18
34
3,
35
49
34
35
73
J5
78
34
3i
35
49
35
36
i3
i5
18
34
3i
35
49
36
37
i3
.5
t8
34
3i
35
49
37
38
i3
i5
18
34
3i
36
5£
3B
39
a
[3
i5
iB
31
35
38
33
36
5o
39
4a
73
i5
iB
^
38
33
36
; 5i
4o
4i
i3
i5
[8
35
38
33
37
5o
4i
4j
i3
.5
iB
36
»9
33
37
5i
43
43
■ 4
16
'9
33
36
3l
34
38
5t
43
44
■ 4
16
12
»3
3a
34
38
53
44
45
a
■ 4
16
>9
i3
37
37
35
39
; 5;
45
46
■ 4
16
'9
33
3r
35
39
53
46
4?
14
16
j3
3.
35
4o
54
47
49
■ 4
16
.3
37
3,
35
4o
55
48
49
1
■ 4
17
ar
34
38
33
36
4i
57
5.)
5j
°
i
14
■ 4
i5
Is
i;
=4
35
38
38
39
37
33
33
37,
37
37
4r
43
43
58
5i
5a
53
i5
35
39
33
38
43
61
53
54
:
16
1?
i3
16
3o
34
39
44
! fi3
54
55
1
76
'9
i6
3o
35
4o
45
'■ S3
55
56
3
16
U
3i
36
4<
46
: 54
56
57
4
33
37
43
48
; 56
57
58
4
17
35
39
33
3a
44
5o
! 68
58
g
?
1
^
^ -
5
5
18
'9
n
»6
3^
37
34
35
39
4o
45
46
5i
5;
i ^
59
6i
5
'9
33
3i
36
41
47
54
( 75
6(
6i
H
3B
33
37
43
49
56
( 78
63
63
TO
U
39
33
39
44
5i
5B
( 3t
63
64
6
7
35
=9
34
40
46
53
60
( w
64
65
6
7
;5
3^
35
4i
48
55
€i
i i?
65
66
6
>6
3i
37
43
5o
58
65
66
67
6
8
1-3
17
33
38
45
53
5o
67
^ ?5
67
68
6
9
34
18
34
40
48
55
63
68
69
70
3
3
5
6
6 -
35
3o
37
36
38
43
44
5o
5i
58
6S
65
73
1 ^
52
7f
4
6
I
33
40
46
55
63
73
83
( )6
71
4
6
S I
3
'9
35
43
49
58
66
76
87
i »4
7>
LogarilAnu of tin in eempvting Compoimd Interat far long ptriodi of lime.
154=539093394
.U0108 4381) 93330
.0016a 55583 8673?
.ooiiS 60617 565o8
.00 J 70 58o33 75935
.oo3a4 5o548 i3t47
.0037835477 3oiJ7
.oo43i 1373783643
.<m485 85346 ao3i9
.00539 5o3i8 66706
.00593 08673 19313
.00^ 60433 49333
,00700 o55B6 03135
.007534417897359
.008067631748033
.008600171761918
,oo9r3 3o6g5 4o47a
.00966 33r66 79379
,01019 39r47 66475
.01073 38653 91773
.01135 3[70[ 37497
.0T178 i83oS 48107
.0133098483 30336
,01383 7334705173
.013363961557963
.01 38900603 38430
.01441 55j35 60603
.01494 03497 93937
.oi546 45435 5833o
.Oi5o8 8io53e4r3o
.oi65i 10367 VS167
.01703 33391 98780
-0175550144 14844
.oiSoo
.01635
'01875 !
!o3i35
!o3375 1
.o35oo!
.o3635 ;
.o375o I
.03875 1
-04^
Log.
>iBo7 6o636 45795
.1859 54884 91658
11911 6390447073
11963 54710 oi3i6
I30i5 4o3i6 38333
13067 '9738 36757
.01173 75876 31799
.0131534596 33711
.013766711957749
.03438 03760 47080
.03479 34333 38763
O353o 58651 64770
.o35Bi 77033 5:
.3633
19367:
)573o s
i349
.03683 95730 91644
.03734 96077 74757
.0378590441 75579
,03836 78836 07061
,038876137736339
,03938 37776 85310
,039890834931355
,o3o39 73oo8 56763
,03090 31766 39399
,o3i4o 84643 5r634
.03191 3i644 61711
.o334i 75788 33769
,0339a 08087 "3366
,03343 37554 86950
.033936130473871
.o344a 79050 a54o3
,08375
,o85oo
.08635
,08750
,08875
:^375
.03493 91 io4 B4167
.0354397381 84548
-o35o3 97894 56733
,o3643 03656 16675
.0369381680 1573c
.03741 64979 i
.03791 43567 i3636
.o384i 14456 41459
.03891 80660 3o37o
.03941 41191 76137
I .03990 96063 74og
I .o4o4o 45189 i4i5
. .04089888808183
• ,o4i3oi6S5i 5833
I '.04188 59314 1010
11.0433785981 3987
.1.04387 07165 8563
I1.04336 33780 3ir3
. .04385 33837 o5883
1 1.04434 3734895107
.!.o44B3 36338 31 ""
I -.04533 39787 8(._,
if.oJSHi 1773978370
I .046300019651969
i'.o4678 77170 44931
.047174867384179
.04776 14718 96603
.0483475318 03974
,04873 3o483 33968
,04931 80336701"
.04970 3456o 531
.o5oi8 63496 75361
.0506697047 4 1385
3,3o358 50939 o4o45 68401
4.6o5i7 01859 t^oji 368o3
6.90775 53789 81137 oSioS
6
9.31034 03719 76181 73607
11.51191 54649 7033S 41008
i3.8i55i 05579 64374 io4io
8
9
16.11809 565o9 58319 78811
re.43068 07439 53365 47114
ao.73336 58369 46411 i56i6 .,„.
■.43439 44819 o335i 83765 11385
..86858 80638 o65o3 6553o 13578
.30188 34457 07755 48395 33868
.73717 79376 i3oo7 3io6o 45167
,17147 34095 16=59 '3835 56446
.60576 68914 19510 96590 67735
1.04006 13733 33763 79355 79034
1.47435 58551 16014 eiiio 9o3i4
1.90865 03371 39366 44666 Qi6o3
Ltngthi of CirctUar Arcs in dieimal partt of RaiitLt.
.01745 33935 19943 39577
.03490 65850 30686 59154
.05335987755983988731
4 1.06961 31700 79773 i83o8
' >8736 64635 9971647885
0471 97551 1965977463
1117 30476 39603 07o38
8 .13969 63401 59546 366i5
9 1.15707 963i6 79489 66193
10 48481 36811 00536
.000009696173633.;
.00001 45444 10433 if
.00001 93935 47144 38i44
.00001 4a4o6 84o55 47680
.00003 90888 ao866 57316
,oooo3 39369 57677 66753
,0ODo3 87850 94488 76188
.00004 36331 3119985834
.00039 "8883 o6665 73160
.ooo5B 17764 17331 44319
.00087 36646 35997 16479
1 16 355a8 34661 88638
145444104333860798
174531935199433958
.oo3o3 61174 60660 o5ii7
,0033i 7io56 69315 77177
.00361 79938 77991 49437
340
TABLES.
Common Logarithms of J^umberB^ carried to thirty places of deeimals.
w.
Log.
a
3
4
5
6
7
8
9
II
13
i3
17
3?
37
4i
o.3f) 1 02 9905663981 loSai
0.4771a iaj47 1966243739
o.6o2o5 99913 37962 39042
0.69897 00043 36018-80478
0.77815 I a5o3 83643 6325©
0.8450980400 14266 8 3071
0.90308 99869 9»943 58564
0.95424 25094 3o324 87459
.o4i3oi685i 5oa25o4o75
.07918 124604762482773
. 1 1 394 33523 o6«36 76020
.33o44 89213 78373 92854
.37875 36009 52828 061 53
.36173 78360 17593 87886
.46339 799t78 98936 08733
.49136 10938 34273 67966
.56830 17340 66994 99080
.61278 38567 19735 49450
37388 94724
5037903355
74777 89449
6361 1 05276
87667 97980
22162 58593
1216684173
oo558o65io
01999 71343
35o56 93704
65o5i §7042
01698 94J28
6333475757
77771 I335l
38467 63969
67041 001 18
845o6 89539
9411849968
43
47
53
61
67
7M
73 I
8?'
89
lOI j
io3
107 I
137'
1 .63346 84555 70686 62640 5o88i 532?9
1 .67209 78679 357 1 7 4644 1 42 1 93 99449
1 .72437 68690 00789 o4563 29922 91627
1.77086 3011642144 190260656384535
1.78532 98350 10767 o3388 57485 13767
1 .82607 48027 00836 43414 9i3i6 29336
I.85I36 83487 19076 38609 38294 35o35
1.86333 28601 20455901074386000470
1 .89762 7091 a 90441 42799 482 1 3 86478
1.91907 80923 76073 9o383 27603 62027
1 .94939 00066 4491 2 7847a 35433 6970 J
1 .98677 1 734a 66a44 85 1 78 436 1 8 n 666
2.00433 13737 8264a 57427 5i88i 78aa3
3.012837224706172 20617 10711 04680
2.03938 37776 86300 64o83 464i 2 39461
2.r3742 64979 4063d 63530 o5i33 07614
3.o63o7 84434 83419 72279 62370 a86o9
2.io38o 37209 55966 86424 69874 ai827
Jiumbers often used in calculation.
ii= Base of Napier's System of Logarithms =3.71828 18284 69046 23636 02874 7i363
i-f-is=o.36787 94412, log.«=r.56570 66180 96748 17284 88710 8io83
Mas com. log. t = Mod. of Com. Sys. of Log. SBO.43429 44819 0826 1 82766 11289 18917
i+Mssi-f. com. log. 1= hyp. log. of 10=2.302586093994046684017991464684
rsRatio of Circum. to Diameter of a Circle =3.14169 a6536 89798 23846 a6433 88280
n-»=o.8i88o 98862, log.=i.5o285 01278 06866 i4564 87817 11709
com. log. ir^o.49714 98726 94188 65485 ia68a 88291
hyp. log. r ^1.14472 98868 49400 1 74 1 4 34278 5i353
»* =9.869604 401089, i-f-ir*s^.ioi8a 11886, ^r ^1.77245 88609
Arc of same length as Radius sss 1 80^+ rss 10800' -4- v =648000'' -f-r
i8o"-hir=67** .296779518, log =1.76812 26824 09172
10800'-*- r=3437' .7467708, log =8.68627 88827 92816
648ooo"-!- »=2o6264" .80626, log =6.8i44a 5i88i 76469
Length of Arc i"=o.ooooo 48481 86811 09685 99868 99141 in parts of Radius,
as:ir-f-648oOO,
I "=o.oocx)o 4848 1 868 II ,
2 "=0.00000 96962 78622,
8"^:o.ooooi 4^444 io43,
4"^o.ooooi 93926 4724,
5"=o.oooo2 42406 84o5,
9"=o.oooo4 36332 8129,
J.00004 84818 6809,
sin ii"=o.oooo5 33296 0490,
sin i2"s=o.oooo6 81776 4170,
sin I ' =0.00029 08882 o46,
sin i2'=o.()o349 o65i4 i5,
i'^z=o.oi745 24064,
sin
sin
sin
sin
sin
sin
sin 16"
og =4 68667 48668 28540 51968
og =4.68557 48668
og ^=4.98660 48624
og =5,16269 6i2i5
og =5.28768 48680
og =5.38464 48710
og =5.68981 78762
og =6.68667 48665
og =5.72696 76618
og =6.76476 61 126
og =6.46872 61 109
og =7.54200 6481 1
og =8.24i85 58i84
log :s2.56258io
log =2.6626978
sin
Tropical Year =365''. 5*. 48". 4/ -588 =865''. 242 2 17466,
Sidereal Year =365^. 6*. 9'". io'.742 =366'. 256874882,
a4* Sol. T. =24* 8" 56* .655336 Sid. T. =24* X 1.00278791, log 1.002 . .". =0.001 iM
24* Sid. T. =24*-8'»56' .90944 Sol. T. =24* X 0.9972696, log 0.997 . . .=9.9988126
Accel- of Stars in Sol. day =8* 56' .555835 Sid. T. =236'.565385, log =2.8789328
Accel- of Stars in Sol. day =8" 56' .90944 Sol. T. =285* .90944, log =2.8727463
Length of sec. pend. in inch, at Lond. 89.18929, Paris 89.1285, New York 89.1012
^= Accel. Force of Gravity at London SS32.1908 Eng. feet, log =i.5o773i5
I cubic inch of water (Bar. 30 inch. Fahr. Therm. 62°,)
=262.468-1-7000 in Avoir, pounds »a52458 in Troy grains, log 3^2.4021893
British imperial gallon =277.274 cubic inchea, log ssa.4429091
/J^
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